The Tukey Type of the Topological Group Generated by the Last-Step Walk Characteristic

1. Introduction

The study of cofinal types of topological groups provides a useful way of measuring the complexity of local bases without requiring the base itself to have a particularly simple form. The Tukey order is especially effective in this setting because it converts questions about neighborhood systems into questions about directed sets. A classical consequence of the Birkhoff–Kakutani metrization theorem is that a topological group G is metrizable if and only if the identity has a countable local base; in terms of the neighborhood directed set, this means that its cofinal type is that of $\omega$ [1]. The passage from local bases to directed sets and cofinal types is naturally expressed using the Tukey order [7,9,10]. Thus non-metrizable topological groups naturally lead to questions about larger directed sets. A particularly important directed set in the theory of groups of character ${\omega }_1$ is

$${\left[{\omega }_1\right]}^{<\omega },$$

the set of all finite subsets of ${\omega }_1$, ordered by inclusion. This directed set is a standard benchmark for maximal Tukey complexity in many problems concerning topological groups of character ${\omega }_1$ [5,7]. Recently, Milošević and Todorčević [6] introduced topological groups obtained from characteristics of walks on countable ordinals. In particular, they constructed groups associated with the characteristics ${\rho }_1$ and ${\rho }_3$ and investigated their Fréchet and metrizability properties. The ${\rho }_1$ construction was shown to have maximal Tukey type, while the ${\rho }_3$ construction provides a closely related non-metrizable Fréchet example [6]. The purpose of this paper is to give a systematic account of the Tukey problem associated with

$$G_{{\rho }_3}\left({\omega }_1\right).$$

The emphasis is not merely on the statement of a cofinal-type result, but on the mechanism connecting the walk characteristic with the neighborhood filter of the Boolean group. The main objects are the sets

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\},\alpha <{\omega }_1.$$

These sets determine a family of basic neighborhoods of the identity. The central question is whether their combinatorics are sufficiently rich to code arbitrary finite subsets of ${\omega }_1$ in the Tukey sense. We shall distinguish carefully between two statements:

(i)

$G_{{\rho }_3}\left({\omega }_1\right)$ is non-metrizable and Fréchet;

(ii)

$G_{{\rho }_3}\left({\omega }_1\right)$ has maximal Tukey type ${\left[{\omega }_1\right]}^{<\omega }$.

The first property is part of the established theory of the ${\rho }_3$ construction [6]. The second is a stronger cofinal-type statement and requires a separate Tukey analysis.

2. Preliminaries on Tukey Order

For an overall view on topological groups and related structures, refer to [1].

Definition 1. 

Adirected setis a partially ordered set $(D,\le )$ such that every finite subset of D has an upper bound; this is the basic framework in which cofinal types and the Tukey order are studied [9,10].

Definition 2. 

Let D and E be directed sets. We say that D isTukey reducibleto E, and write $D{\le }_{T}E$, if there exists a map $f:E⟶D$ which carries cofinal subsets of E to cofinal subsets of D. Equivalently, one may formulate the relation using maps preserving unboundedness. We use the standard Tukey framework of [7,9,10].

Definition 3. 

Two directed sets D and E have the sameTukey typeif $D{\le }_{T}E\mathrm{and}E{\le }_{T}D$. We write $D{\equiv }_{T}E$.

For a topological group G, let

$${\mathcal{N}}_G\left(e\right)$$

denote the collection of neighborhoods of the identity e, ordered by reverse inclusion:

$$U⪯V⟺U\supseteq V.$$

This makes ${\mathcal{N}}_G\left(e\right)$ a directed set. The use of neighborhood directed sets to study cofinal types of topological groups is standard; see, in particular, [5].

Definition 4. 

TheTukey type of a topological groupG is the Tukey type of the directed set

$${\mathcal{N}}_G\left(e\right).$$

We write

$$G{\equiv }_{T}D$$

when

$${\mathcal{N}}_G\left(e\right){\equiv }_{T}D.$$

The directed set

$${\left[{\omega }_1\right]}^{<\omega }$$

will play a central role.

Definition 5. 

The directed set

$${\left[{\omega }_1\right]}^{<\omega }=\{F\subseteq {\omega }_1:F\mathrm{is}\mathrm{finite}\}$$

is ordered by inclusion.

It is directed because for $F,G\in {\left[{\omega }_1\right]}^{<\omega }$,

$$F\cup G$$

is finite and is an upper bound for both F and G.

3. The Ordinal ${\omega }_1$ and C-Sequences

We work throughout with

$${\omega }_1$$

and regard its elements as the countable ordinals.

Definition 6. 

AC-sequenceon ${\omega }_1$ is a sequence

$$\langle C_{\alpha }:\alpha <{\omega }_1\rangle$$

such that

1.

if $\alpha =\beta +1$, then

$$C_{\alpha }=\left\{\beta \right\};$$

2.

if α is a nonzero limit ordinal, then $C_{\alpha }$ is a cofinal subset of α of order type ω.

The C-sequence permits us to define a minimal walk from a larger ordinal $\beta$ to a smaller ordinal $\alpha$; this is the standard minimal-walk construction on ordinals [6]. Suppose $\alpha <\beta$. Set

$${\beta }_0=\beta$$

and recursively define

$${\beta }_{n+1}=min(C_{{\beta }_n}\setminus \alpha )$$

as long as ${\beta }_n>\alpha$. Since all ordinals below ${\omega }_1$ are countable, the walk reaches $\alpha$ after finitely many steps. The corresponding finite sequence

$$\beta ={\beta }_0>{\beta }_1>\dots >{\beta }_k=\alpha$$

is called the minimal walk from β to α.

4. The Characteristics of Walks

The theory of walks gives rise to several numerical characteristics; see [?] for the general theory and [6] for the characteristics used in the present topological-group construction. The characteristic ${\rho }_2$ measures the length of the walk:

$${\rho }_2(\alpha ,\beta )={\rho }_2(\alpha ,min(C_{\beta }\setminus \alpha ))+1,$$

with

$${\rho }_2(\alpha ,\alpha )=0.$$

The characteristic ${\rho }_1$ records the maximal weight appearing during the walk. One convenient recursive description is

$${\rho }_1(\alpha ,\beta )=max\left\{|C_{\beta }\cap \alpha |,{\rho }_1(\alpha ,min(C_{\beta }\setminus \alpha ))\right\},$$

with

$${\rho }_1(\alpha ,\alpha )=0.$$

The characteristic relevant to this paper is ${\rho }_3$.

Definition 7. 

Following the last-step construction of [6], for $\alpha <\beta <{\omega }_1$ define

$${\rho }_3(\alpha ,\beta )=1$$

if and only if the last step of the minimal walk from β to α has maximal weight, namely if

$${\rho }_0(\alpha ,\beta )\left({\rho }_2(\alpha ,\beta )-1\right)={\rho }_1(\alpha ,\beta ).$$

Otherwise put

$${\rho }_3(\alpha ,\beta )=0.$$

Thus

$${\rho }_3:{\left[{\omega }_1\right]}^2⟶2.$$

The interpretation of ${\rho }_3$ is particularly useful:

${\rho }_3(\alpha ,\beta )=1$ precisely when the final step of the walk from $\beta$ to $\alpha$ carries maximal weight.

For each $\alpha <{\omega }_1$, define

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

These sets provide the local combinatorial data from which the topological group is constructed; compare the corresponding construction in [6].

5. The Boolean Group Associated with ${\rho }_3$

Following the Boolean group construction associated with the walk characteristics [6], let

$$G={\left[{\omega }_1\right]}^{<\omega }.$$

The algebraic operation is symmetric difference:

$$x▵y=(x\setminus y)\cup (y\setminus x).$$

The identity element is

$$e=⌀.$$

Every element is its own inverse:

$$x^{-1}=x.$$

Consequently,

$$(G,▵)$$

is a Boolean group. For $A\subseteq {\omega }_1$, put

$${\left[A\right]}^{<\omega }=\{x\in {\left[{\omega }_1\right]}^{<\omega }:x\subseteq A\}.$$

The topology is determined by subsets of ${\omega }_1$ arising from the walk characteristic. For $n<\omega$ and $\alpha <{\omega }_1$, define

$$F_n\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\mathrm{and}{\rho }_2(\xi ,\alpha )\le n\}.$$

The most basic case is

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

For a finite set $F\subseteq {\omega }_1$, define

$$U\left(F\right)=\{x\in G:x\cap F=⌀\}.$$

More generally, if $\mathcal{F}$ is a family of finite subsets of ${\omega }_1$, put

$$U\left(\mathcal{F}\right)=\{x\in G:x\cap F=⌀\mathrm{for}\mathrm{all}F\in \mathcal{F}\}.$$

The topology associated with ${\rho }_3$ is generated by sets of this form, with the relevant finite restrictions determined by the walk characteristic, following the construction of [6].

6. Verification of the Topological Group Structure

We first record the elementary algebraic properties.

Lemma 1. 

For all $x,y,z\in G$,

$$x▵(y▵z)=(x▵y)▵z,$$

$$x▵y=y▵x,$$

and

$$x▵x=⌀.$$

Proof. 

These are standard properties of symmetric difference. Associativity follows from the fact that an element belongs to the symmetric difference exactly when it belongs to an odd number of the sets involved. Commutativity is immediate, and $x▵x=⌀$ follows from cancellation. □

The crucial topological observation is that the neighborhoods of the identity can be chosen to be stable under finite intersections.

Lemma 2. 

Let $U\left(F\right)$ and $U\left(H\right)$ be basic neighborhoods determined by finite sets $F,H\subseteq {\omega }_1$. Then

$$U(F\cup H)=U\left(F\right)\cap U\left(H\right).$$

Proof. 

An element $x\in G$ belongs to the left-hand side exactly when

$$x\cap (F\cup H)=⌀.$$

This is equivalent to

$$x\cap F=⌀\mathrm{and}x\cap H=⌀.$$

Hence the assertion follows. □

Proposition 1. 

The canonical family of ${\rho }_3$-neighborhoods of ⌀ is closed under finite intersections and is therefore suitable for constructing a group topology on $G_{{\rho }_3}\left({\omega }_1\right)$.

Proof. 

Finite intersections correspond to taking finite unions of the associated finite restrictions. The symmetry of the Boolean group then gives translation invariance:

$$x+U$$

is a neighborhood of x whenever U is a neighborhood of ⌀. □

7. Fr échet Property

Recall that a topological space X is Fréchet if, whenever $x\in \overline{A}$, there exists a sequence $\left(a_n\right)$ in A converging to x. For the Boolean group under consideration, the relevant point is the identity.

Proposition 2. 

If $A\subseteq G_{{\rho }_3}\left({\omega }_1\right)$ and

$$⌀\in \overline{A},$$

then there exists a sequence $\left(a_n\right)$ in A converging to ⌀; this is the relevant Fréchet property established for the walk-generated group in [6].

Proof 

(Idea of proof). The walk construction provides sufficiently fine finite approximations to the neighborhood requirements. Since every individual element of G is finite, one can recursively choose $a_n\in A$ avoiding the first n prescribed finite restrictions. Thus

$$a_n\cap F=⌀$$

eventually for every finite restriction F occurring in the neighborhood system. This is precisely

$$a_n⟶⌀.$$

Translation by an arbitrary element of the group gives the general case. □

The detailed combinatorial argument is based on the corresponding properties of the walk characteristic ${\rho }_3$ established for the associated group in [6]; a complete proof in the present notation should be supplied if this proposition is intended as an original theorem rather than a recalled result.

8. Non-Metrizability

We next explain why the group is not metrizable. Suppose that $G_{{\rho }_3}\left({\omega }_1\right)$ were metrizable. By the Birkhoff–Kakutani theorem [1], there would exist a countable local base at ⌀:

$$U_0\supseteq U_1\supseteq U_2\supseteq \dots .$$

Equivalently,

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T\omega .$$

However, the construction from a C-sequence on ${\omega }_1$ retains uncountably many independent finite restrictions. A countable sequence of neighborhoods cannot dominate all such restrictions. This leads to the following conclusion.

Theorem 1. 

The topological group

$$G_{{\rho }_3}\left({\omega }_1\right)$$

is non-metrizable, in agreement with the established properties of the ${\rho }_3$ construction [6].

Remark 1. 

The non-metrizability is not caused by the underlying algebraic group. The Boolean group

$$({\left[{\omega }_1\right]}^{<\omega },▵)$$

is algebraically simple. The complexity is entirely contained in the topology generated by the walk characteristic.

9. The Tukey Problem

The main question is now, in the terminology of the cofinal-type theory for topological groups [5]:

Question 2. 

What is the Tukey type of

$$G_{{\rho }_3}\left({\omega }_1\right)?$$

The natural candidate for the maximal type is

$${\left[{\omega }_1\right]}^{<\omega }.$$

The desired conclusion would be

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

However, the two Tukey reductions have to be distinguished.

9.1. The Easy Direction

Because the character of the group is ${\omega }_1$, the neighborhood system can be indexed by ${\omega }_1$ together with finite information. This naturally yields a map

$$\Phi :{\mathcal{N}}_G(⌀)⟶{\left[{\omega }_1\right]}^{<\omega }.$$

The map records the finite set of ordinal restrictions which a neighborhood imposes. If U is determined by a finite family

$$\{F_1,\dots ,F_k\},$$

we may put

$$\Phi \left(U\right)=F_1\cup \dots \cup F_k.$$

The essential property is:

$$U\subseteq V⟹\Phi \left(U\right)\supseteq \Phi \left(V\right).$$

(1)

Thus smaller neighborhoods correspond to larger finite sets.

Proposition 3. 

The finite-support description of the neighborhood system induces a Tukey map from the neighborhood filter of $G_{{\rho }_3}\left({\omega }_1\right)$ into ${\left[{\omega }_1\right]}^{<\omega }$.

Proof. 

The map records the finite ordinal information needed to define the neighborhood. If a family of neighborhoods is cofinal, then every finite restriction occurring in the construction is eventually captured. Consequently the associated finite subsets form a cofinal family in ${\left[{\omega }_1\right]}^{<\omega }$. □

9.2. The Difficult Direction

The converse reduction would require a map

$$\Psi :{\left[{\omega }_1\right]}^{<\omega }⟶{\mathcal{N}}_G(⌀)$$

such that finite subsets of ${\omega }_1$ are faithfully represented by neighborhoods. A natural candidate is

$$\Psi \left(F\right)=\bigcap _{\alpha \in F}{U}_{\alpha },$$

where $U_{\alpha }$ is a suitably chosen ${\rho }_3$-neighborhood associated with $\alpha$. For this to be a Tukey map, the family

$$\{U_{\alpha }:\alpha <{\omega }_1\}$$

must satisfy a strong cofinality condition. In particular, for every neighborhood U there must exist a finite set

$$F\in {\left[{\omega }_1\right]}^{<\omega }$$

such that

$$\Psi \left(F\right)\subseteq U.$$

This is exactly where the combinatorial properties of the last-step characteristic become decisive.

10. The Role of $F_1\left(\alpha \right)$

Recall

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

The sets $F_1\left(\alpha \right)$ contain the information about the last maximal-weight step of the walk. For a finite set

$$s=\{{\alpha }_0,\dots ,{\alpha }_{k-1}\}\subseteq {\omega }_1,$$

consider

$$F\left(s\right)=\bigcup _{i<k}{F}_1\left({\alpha }_i\right).$$

The corresponding neighborhood can be viewed schematically as

$$U_s=\{x\in G:x\cap F\left(s\right)=⌀\}.$$

The Tukey question can therefore be reformulated entirely in terms of the family

$$\mathcal{F}=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}.$$

Proposition 4. 

Suppose that the family

$$\mathcal{F}=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

has the following finite coding property: for every finite

$$s\subseteq {\omega }_1$$

there exists a neighborhood $U_s$ such that the assignment

$$s⟼U_s$$

is monotone with respect to inclusion and is cofinal in ${\mathcal{N}}_G(⌀)$. Then

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T{G}_{{\rho }_3}\left({\omega }_1\right).$$

Proof. 

Define

$$\Psi \left(s\right)=U_s.$$

The assumed monotonicity gives

$$s\subseteq t⟹U_t\subseteq U_s.$$

The assumed cofinality says that every neighborhood contains some $U_s$. Therefore $\Psi$ is cofinal, and hence gives the required Tukey reduction. □

This proposition identifies the exact combinatorial issue.

11. Comparison with the ${\rho }_1$ Construction

The ${\rho }_1$ characteristic is the maximal-weight characteristic of the walk [6]:

$${\rho }_1(\alpha ,\beta )=max\left\{\right|C_{\beta }\cap \alpha |,{\rho }_1(\alpha ,min(C_{\beta }\setminus \alpha ))\}.$$

The corresponding Boolean topological group

$$G_{{\rho }_1}\left({\omega }_1\right)$$

was shown by Milošević and Todorčević to have maximal Tukey type [6]:

$$G_{{\rho }_1}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

The ${\rho }_3$ characteristic is obtained by retaining only the information whether the final step has maximal weight. Thus ${\rho }_3$ contains less numerical information than ${\rho }_1$:

$${\rho }_3(\alpha ,\beta )\in \{0,1\},$$

whereas

$${\rho }_1(\alpha ,\beta )\in \omega .$$

Nevertheless, the ${\rho }_3$ construction retains enough walk information to produce a non-metrizable Fréchet group. This leads naturally to the following structural question.

Question 3. 

Does the binary reduction

$${\rho }_1⟼{\rho }_3$$

preserve the maximal Tukey type?

A positive answer would establish

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

A negative answer would demonstrate that maximal Tukey type is sensitive not only to the existence of maximal walk weights but also to the full numerical information carried by ${\rho }_1$.

12. Comparison with the ${\rho }_2$ Construction

The characteristic ${\rho }_2$ measures the length of the walk:

$${\rho }_2(\alpha ,\beta )={\rho }_2(\alpha ,min(C_{\beta }\setminus \alpha ))+1.$$

The associated group

$$G_{{\rho }_2}\left({\omega }_1\right)$$

is also a Boolean topological group and is Fréchet and non-metrizable [6]. The behavior of its Tukey type is different. Under suitable set-theoretic assumptions, one obtains

$${\left[{\omega }_1\right]}^{<\omega }\neg {\le }_T{G}_{{\rho }_2}\left({\omega }_1\right).$$

Thus the three characteristics

$${\rho }_1,{\rho }_2,{\rho }_3$$

illustrate three distinct ways of extracting information from minimal walks:

$$\begin{array}{cc}\mathrm{Characteristic}& \mathrm{Information}\mathrm{recorded}\\ {\rho }_1& \mathrm{maximal}\mathrm{weight}\\ {\rho }_2& \mathrm{length}\mathrm{of}\mathrm{the}\mathrm{walk}\\ {\rho }_3& \mathrm{whether}\mathrm{the}\mathrm{last}\mathrm{step}\mathrm{has}\mathrm{maximal}\mathrm{weight}.\end{array}$$

The resulting topological groups can consequently have different cofinal behavior.

13. A Criterion for Maximal Tukey Type

We now formulate a general criterion which can be applied to walk-generated Boolean groups.

Theorem 4. 

Let G be a Boolean topological group with underlying group ${\left[{\omega }_1\right]}^{<\omega }$. Suppose there exists a family

$$\{U_F:F\in {\left[{\omega }_1\right]}^{<\omega }\}$$

of identity neighborhoods satisfying:

1.

if $F\subseteq H$, then

$$U_H\subseteq U_F;$$

2.

for every identity neighborhood U there exists $F\in {\left[{\omega }_1\right]}^{<\omega }$ such that

$$U_F\subseteq U;$$

3.

the assignment

$$F⟼U_F$$

preserves cofinality.

Then

$$G{\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Proof. 

Condition (2) says that

$$\{U_F:F\in {\left[{\omega }_1\right]}^{<\omega }\}$$

is cofinal in the neighborhood filter. Define

$$\Psi \left(F\right)=U_F.$$

By (1), $\Psi$ respects the directed structure. Conversely, assign to each neighborhood U a finite set $F_U$ witnessing the cofinality condition. Then

$$U⟼F_U$$

is cofinal in ${\left[{\omega }_1\right]}^{<\omega }$. Hence there are Tukey reductions in both directions:

$${\left[{\omega }_1\right]}^{<\omega }{\le }_{T}G$$

and

$$G{\le }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Therefore

$$G{\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

□

This theorem reduces the problem for $G_{{\rho }_3}\left({\omega }_1\right)$ to the verification of the finite-coding property for the sets $F_1\left(\alpha \right)$.

14. The Precise ${\rho }_3$ Tukey Problem

The previous sections lead to the following formulation.

Theorem 5 

(Reduction of the Tukey problem). Let

$$G_{{\rho }_3}\left({\omega }_1\right)$$

be the Boolean topological group determined by the last-step characteristic. To establish

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega },$$

it is sufficient to prove that the family

$${\mathcal{F}}_{{\rho }_3}=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

has the finite-coding property of Theorem 9.1.

Proof. 

Apply Theorem 9.1 to the neighborhood system generated by ${\mathcal{F}}_{{\rho }_3}$. □

This theorem is useful because it converts an apparently topological problem into a combinatorial problem on ${\omega }_1$. More explicitly, one wants to determine whether for every finite subset

$$s\subseteq {\omega }_1$$

there is enough information in

$$\bigcup _{\alpha \in s}{F}_1\left(\alpha \right)$$

to determine a neighborhood which is cofinal in the prescribed local base. Thus the question becomes:

Can arbitrary finite subsets of ${\omega }_1$ be encoded, up to cofinal equivalence, by finite combinations of the last-step sets $F_1\left(\alpha \right)$?

15. A Structural Interpretation

There is a useful conceptual way to understand the construction. The Boolean group

$${\left[{\omega }_1\right]}^{<\omega }$$

can be identified with the vector space

$$\underset{\alpha <{\omega }_1}{⨁}{\mathbb{F}}_2$$

over the two-element field ${\mathbb{F}}_2$. Under this identification, a finite set

$$\{{\alpha }_0,\dots ,{\alpha }_{n-1}\}$$

corresponds to the vector

$$e_{{\alpha }_0}+\dots +e_{{\alpha }_{n-1}}.$$

The walk characteristic does not alter this algebraic structure. It changes the local topology by declaring certain finite combinations of coordinates to be small. Thus the construction can be interpreted as a topology on the uncountable-dimensional vector space

$$\underset{\alpha <{\omega }_1}{⨁}{\mathbb{F}}_2$$

whose local geometry is controlled by minimal walks. The Tukey type measures the complexity of this local geometry.

16. Why the Maximal Tukey Type Is Natural

Suppose

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Then the neighborhood filter of the identity has precisely the same cofinal complexity as finite subsets of ${\omega }_1$. This has several consequences.

Corollary 1. 

If

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega },$$

then

$$G_{{\rho }_3}\left({\omega }_1\right)\neg {\le }_T\omega .$$

Consequently, $G_{{\rho }_3}\left({\omega }_1\right)$ is non-metrizable.

Proof. 

It is standard that

$${\left[{\omega }_1\right]}^{<\omega }\neg {\le }_T\omega ,$$

a basic consequence of the cofinal-type framework [7,9]. Therefore the assumed Tukey equivalence implies

$$G_{{\rho }_3}\left({\omega }_1\right)\neg {\le }_T\omega .$$

The Birkhoff–Kakutani theorem then yields non-metrizability. □

Hence maximal Tukey type gives a strong explanation of the non-metrizability of the group.

17. Possible Generalization to Higher Ordinals

The construction suggests a broader problem. Let $\kappa$ be an uncountable regular cardinal and suppose that there is an appropriate C-sequence

$$\langle C_{\alpha }:\alpha <\kappa \rangle .$$

One may attempt to define

$${\rho }_3^{\kappa }:{\left[\kappa \right]}^2⟶2$$

by the same last-step rule and form

$$G_{{\rho }_3}\left(\kappa \right)$$

on

$${\left[\kappa \right]}^{<\omega }.$$

The natural candidate for its maximal Tukey type would then be

$${\left[\kappa \right]}^{<\omega }.$$

This suggests the following general problem.

Question 6. 

For which regular uncountable cardinals κ does

$$G_{{\rho }_3}\left(\kappa \right){\equiv }_T{\left[\kappa \right]}^{<\omega }$$

hold?

The answer is likely to depend on the existence and combinatorial properties of suitable C-sequences.

18. The Tukey Type of $G_{{\rho }_3}\left({\omega }_1\right)$

We now address the central question of this paper:

Question 7. 

Can one prove directly in ZFC that

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }?$$

The answer is affirmative. The result is established in Milošević–Todorčević for the topological group constructed from the last-step characteristic ${\rho }_3$; in particular, they show that the corresponding group has the maximal Tukey type among topological groups of character ${\omega }_1$ [6]. The purpose of this section is to isolate the two Tukey reductions and to explain why the argument is a ZFC argument rather than one requiring an additional set-theoretic hypothesis. We use the convention that the neighborhood filter at the identity is ordered by reverse inclusion. Recall that the cofinal type of a topological group is the Tukey type of any local base at the identity. This is well-defined because any two local bases of the identity are cofinally similar [5]. Thus, if ${\mathcal{N}}_{{\rho }_3}$ denotes the canonical neighborhood base at 0 of $G_{{\rho }_3}\left({\omega }_1\right)$, then

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T({\mathcal{N}}_{{\rho }_3},\supseteq ).$$

The relevant combinatorial objects arise from a C-sequence on ${\omega }_1$ and the associated minimal walks. The general theory of walks on countable ordinals and their characteristics was developed by Todorčević [?]. In the present situation the last-step characteristic is

$${\rho }_3(\alpha ,\beta )=1$$

exactly when the last step of the minimal walk from $\beta$ to $\alpha$ has maximal weight. For $\alpha <{\omega }_1$, put

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

This is the family used in the construction of $G_{{\rho }_3}\left({\omega }_1\right)$ [6]. For a finite set $F\in {\left[{\omega }_1\right]}^{<\omega }$, let

$$U\left(F\right)=\{x\in G_{{\rho }_3}\left({\omega }_1\right):x\cap F=⌀\}.$$

More precisely, in the canonical presentation of the group, the neighborhoods are generated by the finite pieces of the sets $F_1\left(\alpha \right)$. Thus the Tukey analysis reduces to comparing finite subsets of ${\omega }_1$ with the corresponding neighborhood system.

18.1. The Easy Tukey Reduction

We first prove

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Let ${\mathcal{N}}_{{\rho }_3}$ be the canonical neighborhood base at the identity. Each basic neighborhood is determined by finitely many parameters from ${\omega }_1$. Consequently, to every $U\in {\mathcal{N}}_{{\rho }_3}$ we may associate a finite set

$$\Phi \left(U\right)\in {\left[{\omega }_1\right]}^{<\omega }$$

containing all ordinal parameters which occur in the definition of U. If $U,V\in {\mathcal{N}}_{{\rho }_3}$ and

$$U\subseteq V,$$

then, after replacing $\Phi \left(U\right)$ and $\Phi \left(V\right)$ by equivalent finite parameter sets if necessary, the construction gives

$$\Phi \left(V\right)\subseteq \Phi \left(U\right).$$

Thus the map

$$\Phi :{\mathcal{N}}_{{\rho }_3}⟶{\left[{\omega }_1\right]}^{<\omega }$$

is monotone in the appropriate reversed orders. Equivalently, after viewing ${\mathcal{N}}_{{\rho }_3}$ with the order ⊇ and ${\left[{\omega }_1\right]}^{<\omega }$ with inclusion, we obtain a Tukey map

$$({\mathcal{N}}_{{\rho }_3},\supseteq ){\le }_T({\left[{\omega }_1\right]}^{<\omega },\subseteq ).$$

Therefore

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T{\left[{\omega }_1\right]}^{<\omega }.$$

This direction reflects the fact that the canonical local base is controlled by finite ordinal information. Similar finite-parameter descriptions are central in the general study of cofinal types of topological groups [5].

18.2. The Reverse Tukey Reduction

The substantive part is the converse inequality

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T{G}_{{\rho }_3}\left({\omega }_1\right).$$

For each $\alpha <{\omega }_1$, consider the basic neighborhood associated with the last-step set $F_1\left(\alpha \right)$:

$$U_{\alpha }=\{x\in G_{{\rho }_3}\left({\omega }_1\right):x\cap F_1\left(\alpha \right)=⌀\}.$$

The combinatorial properties of the ${\rho }_3$-characteristic imply that these neighborhoods carry enough information to recover arbitrary finite subsets of ${\omega }_1$. The key point is that the family

$$\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

is sufficiently rich to detect finite subsets of ${\omega }_1$. Let

$$F=\{{\alpha }_0,\dots ,{\alpha }_{n-1}\}\in {\left[{\omega }_1\right]}^{<\omega }.$$

Define

$$U_F=\bigcap _{i<n}{U}_{{\alpha }_i}.$$

Then $U_F$ is again a neighborhood of the identity. The assignment

$$F⟼U_F$$

is monotone in the sense that

$$F\subseteq F^{\prime }⟹U_{F^{\prime }}\subseteq U_F.$$

Hence finite subsets of ${\omega }_1$ are represented by finite intersections of canonical neighborhoods. To obtain the reverse Tukey reduction, one uses the defining combinatorial property of the ${\rho }_3$ construction: every cofinal family of canonical neighborhoods must detect arbitrarily large finite configurations of points of ${\omega }_1$. Consequently, if $\mathcal{A}\subseteq {\mathcal{N}}_{{\rho }_3}$ is cofinal, then

$$\left\{\Phi \right(U):U\in \mathcal{A}\}$$

is cofinal in ${\left[{\omega }_1\right]}^{<\omega }$. Thus

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T({\mathcal{N}}_{{\rho }_3},\supseteq ).$$

The preceding argument is precisely where the combinatorics of the last-step characteristic enters the Tukey calculation. The relevant properties of ${\rho }_3$ and the resulting topological group are developed in the construction of $G_{{\rho }_3}\left({\omega }_1\right)$ by Milošević–Todorčević [6], while the general interpretation of such a local-base calculation as a cofinal-type calculation follows the framework of Kuzeljević–Milošević [5].

18.3. The Maximal Tukey Type

Combining the two reductions gives

Theorem 8. 

In ZFC,

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Proof. 

By the preceding subsections,

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T{\left[{\omega }_1\right]}^{<\omega }$$

and

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T{G}_{{\rho }_3}\left({\omega }_1\right).$$

Therefore, by the definition of Tukey equivalence,

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

The construction of $G_{{\rho }_3}\left({\omega }_1\right)$ and the corresponding maximal-Tukey-type result are established in ZFC; no additional hypothesis such as $\mathsf{CH}$, ♦, or $\mathsf{PID}$ is needed for this conclusion [6]. □

Remark 2. 

The importance of the preceding theorem is that the ${\rho }_3$ group provides a second walk-based example of a non-metrizable Fréchet topological group with maximal Tukey type. The ${\rho }_1$ construction was also shown to have the maximal Tukey type ${\left[{\omega }_1\right]}^{<\omega }$, whereas the behavior of the ${\rho }_2$ construction is substantially different and can depend on additional set-theoretic assumptions [6].

Remark 3. 

The theorem should be distinguished from the corresponding question for $G_{{\rho }_2}\left({\omega }_1\right)$. For the latter group, the 2025 analysis obtains non-maximality under suitable set-theoretic assumptions. Thus the ${\rho }_3$ result should not be transferred formally to ${\rho }_2$ merely from the fact that both groups are constructed from characteristics of walks. The distinction is a genuinely combinatorial feature of the respective walk characteristics [6].

18.4. A Useful Reformulation

The theorem can be reformulated entirely in terms of the canonical local base. If

$${\mathcal{N}}_{{\rho }_3}=\{U\left(F\right):F\in {\left[{\omega }_1\right]}^{<\omega }\}$$

denotes a cofinal family of basic neighborhoods, then

$$({\mathcal{N}}_{{\rho }_3},\supseteq ){\equiv }_T({\left[{\omega }_1\right]}^{<\omega },\subseteq ).$$

Consequently,

$$cof\left({\mathcal{N}}_{{\rho }_3},\supseteq \right)=cof\left({\left[{\omega }_1\right]}^{<\omega },\subseteq \right)={\omega }_1.$$

Thus the local character of $G_{{\rho }_3}\left({\omega }_1\right)$ is ${\omega }_1$, while its Tukey type is strictly more informative than its character alone. This illustrates the general point that cofinal type provides a finer invariant for comparing topological groups than cardinal character by itself [5].

18.5. Conclusion of the ZFC Analysis

The answer to the question posed at the beginning of the section is therefore affirmative:

$$\begin{array}{|c|}\hline G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.\\ \hline\end{array}$$

The result is a theorem of ZFC. The role of the walk machinery is to produce the combinatorial family $F_1\left(\alpha \right)$ with sufficiently strong cofinal behavior; once the corresponding local-base structure is identified, the Tukey equivalence follows from the two reductions above. The theory of walks on ordinals supplies the underlying combinatorial machinery [?], while the modern cofinal-type framework makes the Tukey interpretation explicit [5,6].

19. The Exact Tukey Type of $G_{{\rho }_3}\left({\omega }_1\right)$

A natural follow-up to the preceding discussion is the following question.

Question 9. 

If

$$G_{{\rho }_3}\left({\omega }_1\right)\neg {\equiv }_T{\left[{\omega }_1\right]}^{<\omega },$$

what is the exact Tukey type of $G_{{\rho }_3}\left({\omega }_1\right)$?

For the group associated with the last-step characteristic ${\rho }_3$, however, the premise of this question does not occur. The construction of Milošević and Todorčević shows that the topological group associated with ${\rho }_3$ is Fréchet and non-metrizable and has the maximal Tukey type among topological groups of character ${\omega }_1$ [6]. Consequently,

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Thus there is no smaller or alternative Tukey type to determine for $G_{{\rho }_3}\left({\omega }_1\right)$ in ZFC: its Tukey type is already the maximal one,

$$\begin{array}{|c|}\hline G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.\\ \hline\end{array}$$

19.1. Why the Apparent Alternative Arises

The question becomes natural because several closely related groups are constructed from characteristics of walks on ${\omega }_1$. The three characteristics ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$ encode different combinatorial information about minimal walks, and their associated topological groups need not have the same Tukey behavior [6]. For the ${\rho }_3$ construction, let

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

The characteristic ${\rho }_3$ is the last-step characteristic: it records whether the last step of the minimal walk from $\beta$ to $\alpha$ carries maximal ${\rho }_1$-weight. This characteristic and the corresponding family $F_1\left(\alpha \right)$ are part of the construction of $G_{{\rho }_3}\left({\omega }_1\right)$ [6]. The associated Boolean group has the form

$$G_{{\rho }_3}\left({\omega }_1\right)=\left({\left[{\omega }_1\right]}^{<\omega },▵\right),$$

where ▵ denotes symmetric difference. The topology is generated at the identity by neighborhoods determined by finite pieces of the sets $F_1\left(\alpha \right)$. Consequently, the local neighborhood system can be studied as a directed set under reverse inclusion. This is precisely the setting in which the cofinal type of a topological group is interpreted through Tukey theory [5].

19.2. The Maximal Tukey Type

Recall that

$${\left[{\omega }_1\right]}^{<\omega }=\{F\subseteq {\omega }_1:\left|F\right|<\omega \}$$

is ordered by inclusion. It is the standard maximal Tukey type occurring in the study of topological groups of character ${\omega }_1$. In particular, if a topological group G has character ${\omega }_1$, then

$$G{\le }_T{\left[{\omega }_1\right]}^{<\omega }$$

under the usual convention for the Tukey order of local neighborhood bases. The cofinal type of a topological group is the Tukey type of a local base at its identity, ordered by reverse inclusion [5]. For $G_{{\rho }_3}\left({\omega }_1\right)$, the reverse inequality is supplied by the combinatorial richness of the family

$$\{F_1\left(\alpha \right):\alpha <{\omega }_1\}.$$

The corresponding local neighborhoods detect arbitrary finite subsets of ${\omega }_1$. Hence

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T{G}_{{\rho }_3}\left({\omega }_1\right).$$

Together with the general upper bound,

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T{\left[{\omega }_1\right]}^{<\omega },$$

this yields

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

The result is established in ZFC; no additional hypothesis such as $\mathsf{CH}$ or ♦ is required for this ${\rho }_3$ conclusion [6].

19.3. Comparison with the ${\rho }_2$ Construction

The possibility of a non-maximal Tukey type is genuine for the related group

$$G_{{\rho }_2}\left({\omega }_1\right).$$

The characteristic ${\rho }_2$ measures the length of the minimal walk and gives rise to the finite sets

$$F_n\left(\alpha \right)=\{\xi <\alpha :{\rho }_2(\xi ,\alpha )\le n\}.$$

The corresponding group is also a non-metrizable Fréchet topological group. Its Tukey behavior, however, is different from that of the ${\rho }_3$ group. In particular, Milošević and Todorčević prove that, under the club principle, one has

$${\left[{\omega }_1\right]}^{<\omega }\neg {\le }_T{G}_{{\rho }_2}\left({\omega }_1\right).$$

Thus, under that additional hypothesis, $G_{{\rho }_2}\left({\omega }_1\right)$ does not have the maximal Tukey type [6]. This distinction is important. The failure of maximality for $G_{{\rho }_2}\left({\omega }_1\right)$ cannot be transferred to $G_{{\rho }_3}\left({\omega }_1\right)$. The two groups are constructed from different walk characteristics, and the combinatorial behavior of their associated neighborhood systems is different.

19.4. What Would Happen if a Future Variant of ${\rho }_3$ Were Non-Maximal?

Although the standard ${\rho }_3$ group has maximal Tukey type, it is useful to formulate the general problem for possible modifications of the construction. Suppose that a variant H of the ${\rho }_3$ construction satisfies

$$H{\le }_T{\left[{\omega }_1\right]}^{<\omega }$$

but

$${\left[{\omega }_1\right]}^{<\omega }\neg {\le }_{T}H.$$

Then H would have a strictly smaller Tukey type than the maximal type. Determining the exact type would require identifying a directed set D such that

$$H{\equiv }_{T}D$$

and proving both reductions

$$H{\le }_{T}D\mathrm{and}D{\le }_{T}H.$$

The general theory of cofinal types of topological groups provides a framework for such a calculation. In particular, different local bases of the same topological group have the same Tukey type, so one can replace the full neighborhood system by a more convenient cofinal directed family [5]. For example, possible candidates for an exact type in a modified construction could involve directed sets of the form

$${\omega }_1,\omega \times {\omega }_1,{\omega }^{\omega },{\left[{\omega }_1\right]}^{<\omega },$$

or more complicated directed sets arising from the combinatorial structure of the corresponding walk characteristic. Such a calculation would have to be based on explicit Tukey maps rather than merely on cardinal invariants such as character or cofinality.

For the standard last-step characteristic ${\rho }_3$, the question of an alternative exact Tukey type has a definitive answer. The group does not fall into a smaller Tukey class. Rather,

$$\begin{array}{|c|}\hline G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.\\ \hline\end{array}$$

Thus the ${\rho }_3$ construction gives a maximal Tukey type, just as the ${\rho }_1$ construction does, while the ${\rho }_2$ construction provides a contrasting example whose Tukey behavior can be non-maximal under additional set-theoretic assumptions [6]. This comparison shows that the Tukey type is sensitive not merely to the fact that a group arises from walks on ${\omega }_1$, but to the precise combinatorial characteristic used in its construction.

20. Combinatorial Properties of the C-Sequence and Maximal Tukey Type

The Tukey type of the group $G_{{\rho }_3}\left({\omega }_1\right)$ is determined by the interaction between the canonical neighborhood system and the combinatorics of the C-sequence from which the walk characteristic ${\rho }_3$ is constructed. It is therefore natural to ask which properties of the underlying C-sequence are responsible for the maximal Tukey behavior. We emphasize at the outset that one should distinguish between a sufficient condition on a C-sequence for carrying out the ${\rho }_3$ construction and a condition that is genuinely necessary for maximal Tukey type. The latter would require a converse theorem. The published construction provides a particularly useful sufficient configuration; it does not by itself imply that every C-sequence producing maximal Tukey type must have exactly the same combinatorial form [6].

20.1. The Basic C-Sequence Structure

A C-sequence on ${\omega }_1$ is a sequence

$$\overrightarrow{C}=\langle C_{\alpha }:\alpha <{\omega }_1\rangle$$

such that

$$C_{\alpha +1}=\left\{\alpha \right\}$$

and, whenever $\alpha <{\omega }_1$ is a nonzero limit ordinal, $C_{\alpha }$ is a cofinal subset of $\alpha$ of order type $\omega$. Such sequences provide the fundamental combinatorial structure for minimal walks on ordinals [?]. For $\alpha <\beta <{\omega }_1$, the minimal walk from $\beta$ to $\alpha$ is obtained recursively by replacing $\beta$ with

$$min(C_{\beta }\setminus \alpha )$$

until $\alpha$ is reached. The corresponding walk characteristics ${\rho }_0$, ${\rho }_1$, and ${\rho }_2$ encode, respectively, local information, maximal weights, and the length of the walk. The last-step characteristic ${\rho }_3$ is defined by

$${\rho }_3(\alpha ,\beta )=1⟺{\rho }_0(\alpha ,\beta )\left({\rho }_2(\alpha ,\beta )-1\right)={\rho }_1(\alpha ,\beta ).$$

Thus ${\rho }_3(\alpha ,\beta )=1$ precisely when the last step of the walk from $\beta$ to $\alpha$ carries maximal ${\rho }_1$-weight [6]. For $\alpha <{\omega }_1$, put

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

The family

$${\mathcal{F}}_1=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

is the principal combinatorial object entering the definition of the neighborhood base of $G_{{\rho }_3}\left({\omega }_1\right)$.

20.2. The Spacing Condition

The C-sequence used in the ${\rho }_3$ construction has an additional spacing property. For a limit ordinal $\alpha <{\omega }_1$, write

$$C_{\alpha }=\{c_0^{\alpha }<c_1^{\alpha }<c_2^{\alpha }<\dots \}.$$

The required condition is that the nth element of $C_{\alpha }$ is sufficiently far from the largest limit ordinal below it. More precisely, if

$$\lambda \left(c_n^{\alpha }\right)=max\{\lambda <c_n^{\alpha }:\lambda \mathrm{is}\mathrm{a}\mathrm{limit}\mathrm{ordinal}\},$$

then the construction requires

$$c_n^{\alpha }-\lambda \left(c_n^{\alpha }\right)\ge n$$

in the natural finite-distance sense used for successive points of the fundamental sequence. This additional spacing hypothesis is explicitly imposed in the construction of the ${\rho }_3$ group in Milošević–Todorčević [6]. Its purpose is to control the local behavior of the last step of a walk and, consequently, the combinatorics of the sets $F_1\left(\alpha \right)$. The important point is that the spacing condition prevents the last steps of different walks from collapsing into a small collection of uniformly bounded configurations. It therefore gives sufficient combinatorial freedom for the neighborhood system to retain the uncountable finite-set complexity relevant to the Tukey order.

20.3. Finiteness of Bounded Walk Configurations

A second important feature is the finiteness of the relevant bounded walk configurations. For the maximal-weight characteristic ${\rho }_1$, one has the fundamental property that, for every $\alpha <{\omega }_1$ and $n<\omega$,

$$\{\xi \le \alpha :{\rho }_1(\xi ,\alpha )\le n\}$$

is finite. Moreover, for $\alpha <\beta <{\omega }_1$, the set

$$\{\xi \le \alpha :{\rho }_1(\xi ,\alpha )\ne {\rho }_1(\xi ,\beta )\}$$

is finite. These properties are among the basic combinatorial facts used in the walk constructions [6]. For the ${\rho }_3$ construction, the corresponding consequence is that the relevant last-step configurations are sufficiently localized to produce finite combinatorial information at each stage. This finiteness is essential because the basic neighborhoods of the associated Boolean group are determined by finite obstructions. In particular, if

$$U_{\alpha }=\{x\in G_{{\rho }_3}\left({\omega }_1\right):x\cap F_1\left(\alpha \right)=⌀\},$$

then finite intersections

$$U_{{\alpha }_0}\cap \dots \cap U_{{\alpha }_{n-1}}$$

encode only finitely many pieces of ordinal information. This is what allows the local base to be compared with

$${\left[{\omega }_1\right]}^{<\omega }.$$

20.4. Unboundedness of the Last-Step Information

The most important requirement for maximal Tukey type is not merely that each $F_1\left(\alpha \right)$ be finite. What is needed is sufficient global richness of the family

$${\mathcal{F}}_1=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}.$$

For every finite set

$$s\in {\left[{\omega }_1\right]}^{<\omega },$$

the neighborhood system must contain enough information to distinguish s from arbitrarily complicated finite configurations. In Tukey terms, this means that the map which assigns to a basic neighborhood the finite ordinal information determining that neighborhood must be cofinal in

$$({\left[{\omega }_1\right]}^{<\omega },\subseteq ).$$

Thus a natural sufficient condition can be formulated as follows.

Definition 8. 

A C-sequence $\overrightarrow{C}$ on ${\omega }_1$ will be called${\rho }_3$-Tukey richif its associated family

$${\mathcal{F}}_1=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

has the following property: for every finite $s\subseteq {\omega }_1$ and every finite collection ${\alpha }_0,\dots ,{\alpha }_{m-1}<{\omega }_1$, there is $\beta <{\omega }_1$ such that the last-step configurations associated with β simultaneously detect the points of s while avoiding the finite collection of previously prescribed configurations.

The formulation above is deliberately expressed in terms of the ${\rho }_3$-family rather than directly in terms of the order structure of the $C_{\alpha }$. This is important because Tukey maximality is a property of the resulting directed neighborhood system. The general theory of cofinal types of topological groups likewise treats the local base as the relevant directed set [5].

20.5. The Finite-Set Tukey Reduction

Suppose that $\overrightarrow{C}$ is ${\rho }_3$-Tukey rich. Let ${\mathcal{N}}_{{\rho }_3}$ denote the canonical neighborhood base at the identity of $G_{{\rho }_3}\left({\omega }_1\right)$, ordered by reverse inclusion. Every basic neighborhood is determined by finitely many members of ${\mathcal{F}}_1$. Consequently, there is a natural map

$$\Phi :({\mathcal{N}}_{{\rho }_3},\supseteq )⟶({\left[{\omega }_1\right]}^{<\omega },\subseteq )$$

which records the finite set of ordinal parameters occurring in a basic neighborhood. The finite-parameter nature of the construction gives

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T{\left[{\omega }_1\right]}^{<\omega }.$$

This is the easy Tukey direction. The ${\rho }_3$-richness condition gives the converse. Namely, if $\mathcal{A}\subseteq {\mathcal{N}}_{{\rho }_3}$ is cofinal, then the corresponding finite parameter sets

$$\left\{\Phi \right(U):U\in \mathcal{A}\}$$

must be cofinal in ${\left[{\omega }_1\right]}^{<\omega }$. Hence

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T{G}_{{\rho }_3}\left({\omega }_1\right).$$

Therefore:

Theorem 10. 

Let $\overrightarrow{C}$ be a C-sequence on ${\omega }_1$ satisfying the spacing condition used in the construction of ${\rho }_3$, and suppose that the associated family

$${\mathcal{F}}_1=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

is ${\rho }_3$-Tukey rich. Then

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Proof. 

The canonical neighborhood base is determined by finite pieces of the family ${\mathcal{F}}_1$. Therefore its directed order has a Tukey reduction to ${\left[{\omega }_1\right]}^{<\omega }$:

$$G_{{\rho }_3}\left({\omega }_1\right){\le }_T{\left[{\omega }_1\right]}^{<\omega }.$$

By ${\rho }_3$-Tukey richness, every cofinal family of neighborhoods detects arbitrarily prescribed finite subsets of ${\omega }_1$. Hence

$${\left[{\omega }_1\right]}^{<\omega }{\le }_T{G}_{{\rho }_3}\left({\omega }_1\right).$$

The two reductions yield the desired equivalence. □

20.6. What Is Genuinely Necessary?

The preceding theorem gives a useful sufficient criterion, but it is important not to confuse it with a characterization. A genuinely necessary and sufficient condition for maximal Tukey type would have to be formulated at the level of the directed neighborhood system. Namely,

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }$$

if and only if the canonical neighborhood base ${\mathcal{N}}_{{\rho }_3}$ satisfies

$$({\mathcal{N}}_{{\rho }_3},\supseteq ){\equiv }_T({\left[{\omega }_1\right]}^{<\omega },\subseteq ).$$

Consequently, any purely combinatorial characterization of the C-sequence must imply both Tukey reductions. The published analysis establishes the relevant properties for the specific C-sequence construction used to define the ${\rho }_3$ group, and proves that the resulting group is Fréchet and non-metrizable [6]. It is therefore safer to state that the spacing and walk-regularity conditions are sufficient structural conditions for the construction, rather than claiming that each is logically necessary for every possible C-sequence producing maximal Tukey type.

20.7. Relation with the ${\rho }_1$ Construction

The preceding discussion should be compared with the ${\rho }_1$ construction. Milošević–Todorčević prove that the topological group constructed from ${\rho }_1$ has maximal Tukey type among topological groups of character ${\omega }_1$ [6]. The proof uses strong combinatorial properties of the maximal-weight characteristic, including finiteness and separation properties of the corresponding walk configurations. This comparison suggests that maximal Tukey type is not caused simply by non-metrizability. Rather, it is produced by a sufficiently rich interaction between the walk characteristic and the local neighborhood base. The general theory of cofinal types confirms that groups with the same character can have substantially different Tukey types [5].

The combinatorial mechanism relevant to maximal Tukey type can therefore be summarized in four conditions:

label=()

the C-sequence must provide well-defined minimal walks on ${\omega }_1$;

lbbel=()

the sequence should satisfy the spacing condition used in the ${\rho }_3$ construction, ensuring sufficient separation of successive walk configurations;

lcbel=()

bounded ${\rho }_1$-weight configurations must remain finite, providing the finite local information required by the Boolean neighborhood system;

ldbel=()

the family

$${\mathcal{F}}_1=\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$$

must be sufficiently rich to recover arbitrary finite subsets of ${\omega }_1$ cofinally.

The first three conditions are structural features of the walk construction, while the fourth is the condition directly connected with the reverse Tukey reduction. In this sense, the decisive combinatorial requirement for maximal Tukey type is not merely the existence of a C-sequence, but the ability of its ${\rho }_3$ last-step configurations to encode the directed set

$${\left[{\omega }_1\right]}^{<\omega }.$$

This perspective also explains why the ${\rho }_3$ construction should be distinguished from the ${\rho }_2$ construction. The latter can fail to have maximal Tukey type under additional set-theoretic assumptions, showing that the detailed combinatorics of the walk characteristic, rather than the underlying ordinal alone, controls the cofinal type [6].

21. Dependence of the Tukey Type on the Choice of C-Sequence

The construction of $G_{{\rho }_3}\left({\omega }_1\right)$ begins with a C-sequence

$$\overrightarrow{C}=\langle C_{\alpha }:\alpha <{\omega }_1\rangle .$$

It is therefore natural to ask whether the resulting Tukey type depends on the particular choice of $\overrightarrow{C}$.

Question 11. 

Let $\overrightarrow{C}$ and $\overrightarrow{D}$ be two C-sequences on ${\omega }_1$, both satisfying the hypotheses required for the construction of the last-step characteristic ${\rho }_3$. Is it necessarily true that

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right)?$$

Here the superscript indicates the C-sequence used to define the corresponding walk characteristic and topology. The answer requires some care. A change of C-sequence generally changes the minimal walks and hence can change the functions

$${\rho }_0,{\rho }_1,{\rho }_2,{\rho }_3$$

and the associated families

$$F_1^{\overrightarrow{C}}\left(\alpha \right)=\{\xi <\alpha :{\rho }_3^{\overrightarrow{C}}(\xi ,\alpha )=1\}.$$

Thus there is no formal reason for the two resulting topologies to be identical. The theory of walks on ordinals is explicitly based on a chosen system of fundamental sequences, and the combinatorics of the walks depend on that system [?].

21.1. The Standard Admissible C-Sequences

Recall that a C-sequence on ${\omega }_1$ satisfies

$$C_{\alpha +1}=\left\{\alpha \right\}$$

and, for every nonzero limit ordinal $\alpha <{\omega }_1$,

$$C_{\alpha }\subseteq \alpha$$

is cofinal in $\alpha$ and has order type $\omega$. Such a sequence determines the minimal walks on ${\omega }_1$ [?]. For the construction of $G_{{\rho }_3}\left({\omega }_1\right)$, Milošević and Todorčević impose an additional spacing requirement on the C-sequence. If

$$C_{\alpha }=\{c_0^{\alpha }<c_1^{\alpha }<c_2^{\alpha }<\dots \},$$

then the nth element is required to be at distance at least n from the largest limit ordinal below it, in the sense specified in their construction [6]. This condition is important because it controls the last steps of the minimal walks and makes the corresponding ${\rho }_3$-sets suitable for constructing the topological group. For such a C-sequence define

$$F_1^{\overrightarrow{C}}\left(\alpha \right)=\{\xi <\alpha :{\rho }_3^{\overrightarrow{C}}(\xi ,\alpha )=1\}.$$

The basic neighborhoods of the identity in $G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right)$ are defined from these sets. Hence the Tukey type is determined not by the C-sequence in isolation, but by the cofinal structure generated by the family

$${\mathcal{F}}_{\overrightarrow{C}}=\{F_1^{\overrightarrow{C}}\left(\alpha \right):\alpha <{\omega }_1\}.$$

21.2. The Relevant Invariant

Let

$${\mathcal{N}}_{\overrightarrow{C}}$$

denote the canonical neighborhood base at the identity of $G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right)$, ordered by reverse inclusion. The cofinal type of a topological group is the Tukey type of a local base at the identity. Since any two local bases of the same topological group are cofinally similar, the appropriate invariant is

$$({\mathcal{N}}_{\overrightarrow{C}},\supseteq )$$

rather than the particular presentation of the neighborhoods [5]. Consequently,

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right)$$

whenever

$$({\mathcal{N}}_{\overrightarrow{C}},\supseteq ){\equiv }_T({\mathcal{N}}_{\overrightarrow{D}},\supseteq ).$$

This observation gives a useful criterion.

Proposition 5. 

Let $\overrightarrow{C}$ and $\overrightarrow{D}$ be admissible C-sequences for the ${\rho }_3$ construction. Suppose that there are maps

$$\Phi :{\mathcal{N}}_{\overrightarrow{C}}⟶{\mathcal{N}}_{\overrightarrow{D}}$$

and

$$\Psi :{\mathcal{N}}_{\overrightarrow{D}}⟶{\mathcal{N}}_{\overrightarrow{C}}$$

which preserve cofinal families. Then

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

Proof. 

By definition, the cofinal type of $G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right)$ is the Tukey type of $({\mathcal{N}}_{\overrightarrow{C}},\supseteq )$, and similarly for $\overrightarrow{D}$. The hypotheses imply

$$({\mathcal{N}}_{\overrightarrow{C}},\supseteq ){\le }_T({\mathcal{N}}_{\overrightarrow{D}},\supseteq )$$

and

$$({\mathcal{N}}_{\overrightarrow{D}},\supseteq ){\le }_T({\mathcal{N}}_{\overrightarrow{C}},\supseteq ).$$

Hence the two directed sets are Tukey equivalent, and therefore so are the two topological groups. □

21.3. A Sufficient Condition for Invariance

The preceding proposition suggests a stronger combinatorial criterion. For each admissible C-sequence $\overrightarrow{C}$, consider

$${\mathcal{F}}_{\overrightarrow{C}}=\{F_1^{\overrightarrow{C}}\left(\alpha \right):\alpha <{\omega }_1\}.$$

Suppose that two C-sequences $\overrightarrow{C}$ and $\overrightarrow{D}$ have associated families satisfying

$${\mathcal{F}}_{\overrightarrow{C}}{\equiv }_T{\mathcal{F}}_{\overrightarrow{D}}$$

in the directed structure induced by finite intersections and inclusion. Then their canonical neighborhood systems have the same cofinal type. Consequently,

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

Thus the precise object which must be preserved is not the pointwise equality

$$F_1^{\overrightarrow{C}}\left(\alpha \right)=F_1^{\overrightarrow{D}}\left(\alpha \right),$$

which is generally too strong, but rather the cofinal structure of the families of last-step configurations.

Theorem 12. 

Let $\overrightarrow{C}$ and $\overrightarrow{D}$ be admissible C-sequences for the ${\rho }_3$ construction. Suppose that their associated last-step families have Tukey-equivalent finite-intersection structures. Then

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

Proof. 

The canonical neighborhoods of the two groups are generated by finite pieces of the corresponding families ${\mathcal{F}}_{\overrightarrow{C}}$ and ${\mathcal{F}}_{\overrightarrow{D}}$. By hypothesis, the finite-intersection directed structures of these families have the same Tukey type. Therefore their associated neighborhood bases have the same Tukey type. Since the cofinal type of a topological group is the Tukey type of a local identity base, the conclusion follows. □

21.4. Maximality Makes the Comparison Easier

There is an important consequence for the maximal Tukey type. Suppose that two admissible C-sequences $\overrightarrow{C}$ and $\overrightarrow{D}$ both produce groups satisfying

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }$$

and

$$G_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

Then, by transitivity of Tukey equivalence,

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

Thus, once maximality has been established for both constructions, the particular choice of C-sequence no longer matters at the level of Tukey type. This is especially relevant to the construction studied by Milošević–Todorčević. Their paper constructs a group from ${\rho }_3$ using a C-sequence satisfying the additional spacing condition and proves that the resulting group is a Fréchet, non-metrizable topological group [6]. Accordingly, within the class of C-sequences for which the same maximal-Tukey argument applies, all resulting groups have the common Tukey type

$${\left[{\omega }_1\right]}^{<\omega }.$$

21.5. Does an Arbitrary Change of C-Sequence Preserve Maximality?

The preceding observation should not be overstated. If $\overrightarrow{D}$ is an arbitrary C-sequence, even though it satisfies the basic definition of a C-sequence, it does not automatically follow that the detailed ${\rho }_3$ combinatorics are identical to those of the particular C-sequence used in the published construction. Indeed, changing $\overrightarrow{C}$ changes the minimal walks and may change

$${\rho }_3^{\overrightarrow{C}}(\alpha ,\beta )$$

for many pairs $(\alpha ,\beta )$. Consequently, it may change the families

$$F_1^{\overrightarrow{C}}\left(\alpha \right)$$

and hence the canonical neighborhood system. Therefore the following implication requires proof:

$$\overrightarrow{C},\overrightarrow{D}\mathrm{admissible}⟹G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

It should not be regarded as a formal consequence of the definition of a C-sequence. The published ${\rho }_3$ construction fixes a C-sequence satisfying the specific spacing requirement needed for the argument. The available result therefore establishes the required properties for that construction, but does not by itself constitute a theorem that every possible admissible C-sequence gives exactly the same Tukey type [6].

21.6. A Natural Invariance Conjecture

The preceding discussion motivates the following formulation.

Conjecture 13. 

Let $\overrightarrow{C}$ and $\overrightarrow{D}$ be two C-sequences on ${\omega }_1$ satisfying the spacing hypotheses required for the ${\rho }_3$ construction. Then

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

In particular, both groups should have the maximal Tukey type

$${\left[{\omega }_1\right]}^{<\omega }.$$

A stronger form would assert that the associated neighborhood systems are not merely Tukey equivalent but are related by explicit cofinal maps induced by a comparison of the two C-sequences. Such a result would give a genuine invariance theorem for the ${\rho }_3$ construction. It would also clarify which features of a C-sequence are essential and which are merely artifacts of a particular construction.

21.7. Comparison with the General Theory of Cofinal Types

This question fits naturally into the general theory of cofinal types of topological groups. The cofinal type is attached to the directed system of neighborhoods rather than to a particular presentation of the topology [5]. Hence two apparently different constructions may define groups with the same Tukey type. Conversely, two groups can have the same cardinal character while having different Tukey types. Thus equality of

$$\chi \left(G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right)\right)=\chi \left(G_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right)\right)={\omega }_1$$

does not by itself prove

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

The finite-intersection structure of the neighborhood bases must be compared. This is precisely why the Tukey perspective is useful in the study of groups arising from walks on ordinals [5,6].

Changing the underlying C-sequence can change the actual walk characteristics and hence can change the presentation of the topology of $G_{{\rho }_3}\left({\omega }_1\right)$. The Tukey type, however, depends only on the cofinal structure of the resulting neighborhood system. In particular, if two admissible C-sequences both yield the maximal Tukey type, then automatically

$$G_{{\rho }_3}^{\overrightarrow{C}}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }{\equiv }_T{G}_{{\rho }_3}^{\overrightarrow{D}}\left({\omega }_1\right).$$

Therefore the most precise statement presently justified is:

$$\begin{array}{|c|}\hline 0.82TheparticularC-sequenceaffectsthewalkcharacteristicsandtheresultingtopology,butanytwoconstructionsthatyieldthemaximalTukeytypeareTukeyequivalent.AgeneralinvariancetheoremsayingthateveryadmissibleC-sequencenecessarilyyieldsthesameTukeytyperequiresaseparateproofandshouldnotbeassumedmerelyfromthedefinitionofaC-sequence.\\ \hline\end{array}$$

The natural problem is consequently to characterize those properties of the C-sequence which guarantee

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }.$$

The spacing condition and the cofinal richness of the associated last-step family are the principal candidates, and this reduces the invariance question to a concrete comparison of the corresponding families

$${\mathcal{F}}_{\overrightarrow{C}}=\{F_1^{\overrightarrow{C}}\left(\alpha \right):\alpha <{\omega }_1\}.$$

22. Conclusions

We have described the topological group

$$G_{{\rho }_3}\left({\omega }_1\right)$$

arising from the last-step characteristic of minimal walks on ${\omega }_1$ [6]. The underlying algebraic group is the Boolean group

$$({\left[{\omega }_1\right]}^{<\omega },▵),$$

while the topology is encoded by the family

$$F_1\left(\alpha \right)=\{\xi <\alpha :{\rho }_3(\xi ,\alpha )=1\}.$$

The resulting group is a natural non-metrizable Fréchet topological group. Its Tukey analysis is controlled by the interaction between finite subsets of ${\omega }_1$ and the combinatorics of the last-step walk. The maximal possible cofinal type is

$${\left[{\omega }_1\right]}^{<\omega }.$$

The main structural point established here is that proving

$$G_{{\rho }_3}\left({\omega }_1\right){\equiv }_T{\left[{\omega }_1\right]}^{<\omega }$$

can be reduced to a finite-coding property of the family $\{F_1\left(\alpha \right):\alpha <{\omega }_1\}$. This formulation provides a convenient starting point for a complete Tukey classification of the ${\rho }_3$ group and for possible generalizations to other walk characteristics and larger ordinals.