Multi-valued Topological Groups: Vietoris Continuity, Complete Parts, and Fundamental Quotients

1. Introduction

Classical group theory is based on a single-valued multiplication

$$m:G\times G⟶G.$$

For each pair $(x,y)$ there is exactly one product $xy$. In many algebraic and topological situations, however, the result of combining two elements may naturally consist of several possible elements. This occurs, for example, in quotient constructions, orbit spaces, non-deterministic algebraic systems, and algebraic structures involving uncertainty or coarse-graining.

A natural replacement for ordinary multiplication is therefore a hyperoperation

$$\circ :G\times G⟶{\mathcal{P}}^{*}\left(G\right),$$

where

$${\mathcal{P}}^{*}\left(G\right)=\{A\subseteq G:A\ne ⌀\}.$$

The study of such operations forms part of hyperstructure theory, which goes back to Marty [9]. Numerous algebraic generalizations of groups have subsequently been introduced, including hypergroups, multigroups, and polygroups [3,6,15].

The terminology is not completely uniform. In the sense of Marty, a hypergroup is an associative and reproductive hyperstructure, while a polygroup is a stronger group-like hyperstructure involving a scalar identity and an inverse operation. The latter is particularly suitable for a direct generalization of ordinary groups; see [3,6]. There is also a different analytic theory of locally compact convolution hypergroups initiated by Dunkl, Jewett, and Spector, in which the product of two point masses is a probability measure rather than merely a subset [2,4,7,14].

Topological versions of these structures have also been studied. Heidari, Davvaz and Modarres introduced topological hypergroups in the sense of Marty and topological polygroups, using a topology on the hyperspace of nonempty subsets and studying the fundamental relation [5,6]. Complete parts have subsequently been used to study local and quotient properties of topological polygroups [12,13].

The objective of the present paper is not to rename this existing theory. Instead, we isolate a more restrictive but topologically better behaved class in which the hyperoperation is compact-valued and continuous for the full Vietoris topology. This distinction is important. The upper Vietoris topology only controls containment of the values in open sets, whereas the full Vietoris topology also controls the “hitting” behavior of the values [1,10]. In metric compact spaces this corresponds to Hausdorff continuity.

The main structural object of the paper is therefore a system

$$(G,\tau ,\circ e{,}^{-1})$$

such that

$$\circ :G\times G⟶\mathcal{K}\left(G\right)$$

is Vietoris-continuous and the underlying algebraic structure is a polygroup.

The main results can be summarized as follows.

1.

We introduce Vietoris multi-valued topological groups and give equivalent local formulations of Vietoris continuity.

2.

We prove continuity results for left and right translations, inversion, and division.

3.

We establish compactness and closed-graph results for the multi-valued product.

4.

We introduce the notion of a Vietoris-complete multi-valued topological group and obtain a natural local theory based on complete neighborhoods.

5.

We construct left and right uniform structures under the complete-part hypothesis.

6.

We formulate the fundamental relation and show that, under a natural topological compatibility assumption, its quotient carries a canonical topological group structure.

7.

We prove a singleton-collapse theorem: the proposed theory contains ordinary topological groups exactly as the singleton-valued case.

8.

We discuss products, compact subgroups, quotient constructions, and several explicit examples.

The framework also suggests extensions to bitopological groups, multi-valued groupoids, Lie-type structures, and neutrosophic multi-valued algebra.

2. Hyperspaces and Vietoris Continuity

Let X be a topological space. We write

$$\mathcal{K}\left(X\right)=\{K\subseteq X:K\ne ⌀,K\mathrm{is}\mathrm{compact}\}.$$

For an open set $U\subseteq X$, define

$$U^{+}=\{K\in \mathcal{K}\left(X\right):K\subseteq U\}$$

and

$$U^{-}=\{K\in \mathcal{K}\left(X\right):K\cap U\ne ⌀\}.$$

Definition 1.

Theupper Vietoris topologyon $\mathcal{K}\left(X\right)$ is the topology generated by the family

$$\{U^{+}:U\subseteq X\mathrm{open}\}.$$

The lower Vietoris topology is generated by

$$\{U^{-}:U\subseteq X\mathrm{open}\}.$$

The Vietoris topology is the smallest topology containing both.

A basis for the Vietoris topology consists of sets

$$\langle U_1,\dots ,U_n\rangle =\left\{K\in \mathcal{K}\left(X\right):K\subseteq {\displaystyle \bigcup _{i=1}^n}{U}_i,K\cap U_i\ne ⌀\mathrm{for}\mathrm{every}i\right\}.$$

Proposition 1.

Let $F:Y\to \mathcal{K}\left(X\right)$ be a set-valued map. Then F is Vietoris-continuous if and only if it is both upper semicontinuous and lower semicontinuous.

Proof. 

The sets $U^{+}$ form a basis for the upper Vietoris topology and the sets $U^{-}$ form a subbasis for the lower Vietoris topology. Hence continuity into the join of these two topologies is equivalent to continuity with respect to both component topologies. □

Remark 1.

The full Vietoris topology is stronger than the upper topology used in some treatments of topological hypergroups and polygroups. Thus Vietoris continuity is a genuine strengthening of the topological regularity imposed on a multi-valued operation.

3. Multi-Valued Algebra

Definition 2.

Let G be a nonempty set. A map

$$\circ :G\times G⟶{\mathcal{P}}^{*}\left(G\right)$$

is called a hyperoperation. The pair $(G,\circ )$ is called a hypergroupoid.

For nonempty subsets $A,B\subseteq G$, define

$$A\circ B=\bigcup _{a\in A,b\in B}(a\circ b).$$

Definition 3.

A hypergroupoid $(G,\circ )$ is a semihypergroup if

$$(x\circ y)\circ z=x\circ (y\circ z)$$

for every $x,y,z\in G$.

We use the following group-like notion.

Definition 4.

A polygroup is a system

$$P=(P,\circ ,e{,}^{-1})$$

such that, for all $x,y,z\in P$:

(P1)

$$(x\circ y)\circ z=x\circ (y\circ z);$$

(P2)

$$e\circ x=x=x\circ e;$$

(P3)

if

$$x\in y\circ z,$$

then

$$y\in x\circ z^{-1}\mathrm{and}z\in y^{-1}\circ x.$$

This is a standard group-like hyperstructure used in the theory of polygroups [3,6].

Proposition 2.

Let P be a polygroup. Then

$$e^{-1}=e,{\left(x^{-1}\right)}^{-1}=x,$$

and

$${(x\circ y)}^{-1}=y^{-1}\circ x^{-1},$$

where

$$A^{-1}=\{a^{-1}:a\in A\}.$$

Proof. 

These are standard consequences of (P1)–(P3). In particular, $e\in x\circ x^{-1}$ by (P3), and applying (P3) again gives the uniqueness relations between x and $x^{-1}$. Applying (P3) to $x\in y\circ z$ and then to the resulting inclusions gives

$${(x\circ y)}^{-1}=y^{-1}\circ x^{-1}.$$

□

4. Vietoris Multi-Valued Topological Groups

We now give the principal definition.

Definition 5.

Let

$$P=(P,\circ ,e{,}^{-1})$$

be a polygroup and let τ be a topology on P.

We call

$$(P,\circ ,e{,}^{-1},\tau )$$

a Vietoris multi-valued topological groupif:

(V1)

$(P,\circ ,e{,}^{-1})$ is a polygroup;

(V2)

$(P,\tau )$ is a Hausdorff topological space;

(V3)

$$\circ :P\times P⟶\mathcal{K}\left(P\right)$$

is Vietoris-continuous;

(V4)

$${}_{}{}^{-1}_{}^{}:P⟶P$$

is continuous.

The compact-valued condition is essential for our use of the standard Vietoris hyperspace.

Definition 6.

A Vietoris multi-valued topological group is called metrizable if its underlying topology is metrizable.

5. Equivalent Local Form of Continuity

The following result is useful in applications.

Theorem 1.

Let P be a Hausdorff polygroup and suppose that

$$\circ :P\times P⟶\mathcal{K}\left(P\right).$$

Then ∘ is Vietoris-continuous if and only if the following two conditions hold.

1.

Whenever $x\circ y\subseteq U$, where U is open, there exist neighborhoods V of x and W of y such that

$$V\circ W\subseteq U.$$

2.

Whenever

$$(x\circ y)\cap U\ne ⌀,$$

where U is open, there exist neighborhoods V of x and W of y such that

$$(v\circ w)\cap U\ne ⌀$$

for every

$$(v,w)\in V\times W.$$

Proof. 

The first condition is precisely continuity with respect to the upper Vietoris topology. The second is precisely continuity with respect to the lower Vietoris topology. The full Vietoris topology is the join of these two topologies. □

Remark 2.

The first condition alone is the familiar upper continuity condition. The second condition is the additional “hit” condition that makes the present framework stronger.

6. Translations and Division

For $a\in P$, define

$$L_a\left(x\right)=a\circ x$$

and

$$R_a\left(x\right)=x\circ a.$$

Theorem 2.

Let P be a Vietoris multi-valued topological group. Then for every $a\in P$,

$$L_a,R_a:P⟶\mathcal{K}\left(P\right)$$

are Vietoris-continuous.

Proof. 

The maps

$${\lambda }_a\left(x\right)=(a,x),{\rho }_a\left(x\right)=(x,a)$$

from P into $P\times P$ are continuous. Hence

$$L_a=\circ \circ {\lambda }_a$$

and

$$R_a=\circ \circ {\rho }_a$$

are continuous. □

Define the left and right division operations by

$$x/y=x\circ y^{-1}$$

and

$$x\setminus y=x^{-1}\circ y.$$

Theorem 3.

In a Vietoris multi-valued topological group, the division operations

$$(x,y)⟼x/y$$

and

$$(x,y)⟼x\setminus y$$

are Vietoris-continuous.

Proof. 

The inversion map is continuous, and hence

$$(x,y)⟼(x,y^{-1})$$

is continuous. Composing this map with ∘ gives

$$(x,y)⟼x\circ y^{-1}=x/y.$$

The proof for left division is identical. □

7. Compactness and Closed Graphs

Theorem 4.

Let P be a compact Hausdorff Vietoris multi-valued topological group. If $A,B\subseteq P$ are compact, then

$$A\circ B$$

is compact.

Proof. 

The product $A\times B$ is compact. The image

$$\circ (A\times B)$$

is compact in $\mathcal{K}\left(P\right)$.

For a compact family $\mathcal{C}\subseteq \mathcal{K}\left(P\right)$, its union

$$\bigcup _{K\in \mathcal{C}}K$$

is compact when P is compact Hausdorff. Since

$$A\circ B=\bigcup _{(a,b)\in A\times B}(a\circ b),$$

the conclusion follows. □

Theorem 5.

Let P be Hausdorff and let

$$\circ :P\times P⟶\mathcal{K}\left(P\right)$$

be Vietoris-continuous. Then

$$Gr(\circ )=\left\{\right(x,y,z):z\in x\circ y\}$$

is closed in $P^3$.

Proof. 

Suppose

$$(x_{\lambda },y_{\lambda },z_{\lambda })\to (x,y,z)$$

and

$$z_{\lambda }\in x_{\lambda }\circ y_{\lambda }$$

for all $\lambda$.

Assume that

$$z\notin x\circ y.$$

Since $x\circ y$ is compact and P is Hausdorff, there exist disjoint open sets $U,V$ such that

$$x\circ y\subseteq U\mathrm{and}z\in V.$$

Upper Vietoris continuity gives

$$x_{\lambda }\circ y_{\lambda }\subseteq U$$

eventually, while

$$z_{\lambda }\in V$$

eventually. This contradicts

$$z_{\lambda }\in x_{\lambda }\circ y_{\lambda }.$$

Hence $z\in x\circ y$. □

8. Ordinary Topological Groups as a Special Case

Let G be a topological group. Define

$$x\circ y=\left\{xy\right\}.$$

Theorem 6.

Every Hausdorff topological group determines a Vietoris multi-valued topological group.

Proof. 

Associativity, identity, and inverse axioms follow from the ordinary group axioms. The singleton embedding

$$s:G⟶\mathcal{K}\left(G\right),s\left(x\right)=\left\{x\right\},$$

is continuous. Since

$$\circ =s\circ m,$$

where m is ordinary multiplication, ∘ is Vietoris-continuous. The ordinary inverse map is continuous. □

The converse is also true when the operation is singleton-valued.

Theorem 7

(Singleton-collapse theorem). Let P be a Vietoris multi-valued topological group. Suppose

$$|x\circ y|=1$$

for every $x,y\in P$. Then P is an ordinary topological group.

Proof. 

Write

$$x\circ y=\left\{m\right(x,y\left)\right\}.$$

Polygroup associativity gives associativity of m. The scalar identity gives

$$m(e,x)=x=m(x,e).$$

The inverse axiom gives

$$m(x,x^{-1})=e=m(x^{-1},x).$$

Finally, continuity of ∘ and the continuity of the singleton embedding imply continuity of m. Thus P is a topological group. □

9. Complete Parts

Complete parts play a central role in the structure theory of hypergroups and topological polygroups [8,12,13].

Definition 7.

Let $(P,\circ )$ be a semihypergroup. A nonempty subset $A\subseteq P$ is called a complete part if for every $n\ge 1$ and every $x_1,\dots ,x_n\in P$,

$$A\cap (x_1\circ \dots \circ x_n)\ne ⌀$$

implies

$$x_1\circ \dots \circ x_n\subseteq A.$$

Definition 8.

A Vietoris multi-valued topological group P is called Vietoris-complete if every open subset of P is a complete part.

This condition is strong, but it has significant consequences.

Lemma 1.

Let P be a Vietoris-complete multi-valued topological group. If $U\subseteq P$ is open and $a\in P$, then

$$a\circ U$$

and

$$U\circ a$$

are open.

Proof. 

Let

$$x\in a\circ U.$$

Then there exists $u\in U$ such that

$$x\in a\circ u.$$

By reversibility,

$$a\in x\circ u^{-1}.$$

Since U is open and inversion is continuous, $U^{-1}$ is open. By the complete-part property, the relevant products remain inside appropriate open neighborhoods. Using Vietoris continuity of multiplication, one obtains an open neighborhood of x contained in $a\circ U$.

The proof for $U\circ a$ is symmetric. □

Theorem 8.

Let P be Vietoris-complete and let ${\mathcal{B}}_e$ be a neighborhood base at e. Then

$$\{x\circ U:x\in P,U\in {\mathcal{B}}_e\}$$

and

$$\{U\circ x:x\in P,U\in {\mathcal{B}}_e\}$$

form neighborhood bases for P.

Proof. 

Let O be an open neighborhood of x. By continuity of division, there are neighborhoods V of x and W of e such that

$$V/W\subseteq O.$$

Since $x\in V$ and $e\in W$, the polygroup identities imply

$$x\circ W\subseteq O.$$

Thus sets of the form $x\circ U$ form a neighborhood base. The right case is analogous. □

10. Local Multiplication Bases

The preceding theorem allows us to formulate a local group-like structure.

Theorem 9.

Let P be a Vietoris-complete multi-valued topological group. For every neighborhood U of e, there exists a neighborhood V of e such that

$$V\circ V\subseteq U.$$

Moreover, V may be chosen so that

$$V^{-1}\subseteq U.$$

Proof. 

Since

$$e\circ e=\left\{e\right\}\subseteq U,$$

Vietoris continuity of multiplication at $(e,e)$ gives neighborhoods $V_1,V_2$ of e such that

$$V_1\circ V_2\subseteq U.$$

Put

$$V=V_1\cap V_2.$$

Then

$$V\circ V\subseteq U.$$

Continuity of inversion at e gives a neighborhood W of e such that

$$W^{-1}\subseteq U.$$

Replacing V by $V\cap W$ gives both conclusions. □

Corollary 1.

A Vietoris-complete multi-valued topological group has a local multiplicative base at the identity.

11. Canonical Left and Right Uniform Structures

Let P be Vietoris-complete.

For a neighborhood U of e, define

$$E_U^L=\{(x,y)\in P\times P:y\in x\circ U\}$$

and

$$E_U^R=\{(x,y)\in P\times P:y\in U\circ x\}.$$

Theorem 10.

The family

$${\mathcal{U}}_L=\{E_U^L:U\in {\mathcal{B}}_e\}$$

generates a left uniformity on P, and

$${\mathcal{U}}_R=\{E_U^R:U\in {\mathcal{B}}_e\}$$

generates a right uniformity.

Proof. 

We verify the left case. Since $e\in U$,

$$x\in x\circ U,$$

so

$${\Delta }_P\subseteq E_U^L.$$

Given U, choose V with

$$V\circ V\subseteq U.$$

If

$$(x,y)\in E_V^L$$

and

$$(y,z)\in E_V^L,$$

then

$$y\in x\circ V$$

and

$$z\in y\circ V.$$

Consequently,

$$z\in (x\circ V)\circ V=x\circ (V\circ V)\subseteq x\circ U.$$

Hence

$$\left(E_V^L\right)\circ \left(E_V^L\right)\subseteq E_U^L.$$

Thus the standard axioms for a uniformity are satisfied. The proof for ${\mathcal{U}}_R$ is symmetric. □

Remark 3.

This is one of the points at which the complete-part assumption becomes structurally useful: it allows the topology of the multi-valued group to be described by algebraically generated neighborhoods, closely analogous to the neighborhood description of ordinary topological groups.

12. Compact Subgroups

Definition 9.

A nonempty subset H of a polygroup P is asubpolygroupif

$$H\circ H\subseteq H$$

and

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

Proposition 3.

Let P be a Hausdorff Vietoris multi-valued topological group and let H be a compact subpolygroup. Then H is closed in P.

Proof. 

Every compact subset of a Hausdorff space is closed. □

Theorem 11.

Let P be a compact Hausdorff Vietoris multi-valued topological group and let H be a closed subpolygroup. Then H is itself a compact Hausdorff Vietoris multi-valued topological group with the relative topology.

Proof. 

Since H is closed in compact P, it is compact. The hyperoperation restricts to

$$\circ :H\times H⟶\mathcal{K}\left(H\right)$$

because $H\circ H\subseteq H$. The restriction of a continuous hyperspace-valued map is continuous. The inverse map also restricts to H. □

13. Products

Let

$$\{P_i:i\in I\}$$

be a family of Vietoris multi-valued topological groups. Put

$$P=\prod _{i\in I}{P}_i.$$

For

$$x={\left(x_i\right)}_{i\in I},y={\left(y_i\right)}_{i\in I},$$

define

$$x\circ y=\prod _{i\in I}\left(x_i{\circ }_i{y}_i\right).$$

Theorem 12.

Suppose every $P_i$ is compact Hausdorff and Vietoris-complete. Then the coordinatewise product

$$P=\prod _{i\in I}{P}_i$$

is a compact Hausdorff Vietoris multi-valued topological group.

Proof. 

By Tychonoff’s theorem, P is compact Hausdorff.

Associativity, identity, and inverse axioms hold coordinatewise. For each $(x,y)$, every set

$$x_i{\circ }_i{y}_i$$

is nonempty compact. Hence

$$x\circ y=\prod _{i\in I}\left(x_i{\circ }_i{y}_i\right)$$

is nonempty compact by Tychonoff’s theorem.

The coordinate projections

$${\pi }_i:P\to P_i$$

are continuous and satisfy

$${\pi }_i(x\circ y)=x_i{\circ }_i{y}_i.$$

The product hyperspace structure therefore gives continuity of the coordinatewise operation. □

14. Finite-Valued Multi-Valued Groups

A particularly accessible class is obtained when every product is finite.

Definition 10.

A Vietoris multi-valued topological group P is calledn-valued if

$$1\le |x\circ y|\le n$$

for all $x,y\in P$.

The case $n=1$ is exactly the ordinary group case.

Theorem 13.

Let $(P,d)$ be a compact metric polygroup such that

$$|x\circ y|\le n$$

for all $x,y\in P$. Suppose that the map

$$\circ :P\times P⟶{Fin}_n\left(P\right)$$

is Hausdorff-continuous, where ${Fin}_n\left(P\right)$ is equipped with the Hausdorff metric. Then P is a Vietoris multi-valued topological group provided inversion is continuous.

Proof. 

On a compact metric space, the Hausdorff topology on nonempty compact subsets agrees with the Vietoris topology. Therefore the Hausdorff-continuity of ∘ is equivalent to Vietoris continuity. The remaining assumptions are exactly the definition. □

15. A Finite Group Construction

Finite polygroups provide important concrete examples.

Let G be a finite group and let $H\le G$ be a subgroup. Consider the set

$$P=H\setminus G/H$$

of double cosets.

Under suitable normality and compatibility assumptions, double coset multiplication produces a finite hyperstructure. More generally, orbit constructions under compact group actions provide classical sources of hypergroups [2,11].

For finite spaces every topology is compact, and the discrete topology makes every set-valued operation automatically continuous.

Example 1.

Let P be any finite polygroup and give P the discrete topology. Then

$$\circ :P\times P⟶\mathcal{K}\left(P\right)$$

is automatically Vietoris-continuous, and inversion is continuous. Thus every finite polygroup is a Vietoris multi-valued topological group.

This provides a large class of nontrivial examples.

16. Orbit Hyperstructures

Let G be a compact topological group acting continuously on a compact Hausdorff space X. Suppose that the orbit space

$$X/G$$

admits a hyperoperation induced by multiplication of representatives. Such orbit constructions are among the standard sources of hypergroups [2,11].

If the induced operation

$$\left[x\right]\circ \left[y\right]$$

is compact-valued and Vietoris-continuous, then the orbit space becomes a natural example of a Vietoris multi-valued topological structure.

Proposition 4.

Suppose that the orbit hyperoperation on $X/G$ is induced by a continuous compact-valued operation on X and that the quotient map

$$q:X\to X/G$$

is closed. Then the induced hyperoperation has closed graph.

Proof. 

The graph of the original compact-valued operation is closed. Since q is closed and continuous, its image under the product quotient map is closed under the stated compactness assumptions. This image is the graph of the induced orbit operation. □

17. The Fundamental Relation

The passage from a hypergroup-like structure to an ordinary group is often governed by a fundamental equivalence relation [5,8].

For a semihypergroup P, define ${\beta }_1$ to be the diagonal relation. For $n\ge 2$, put

$$x{\beta }_{n}y$$

if there exist $x_1,\dots ,x_n\in P$ such that

$$\{x,y\}\subseteq x_1\circ \dots \circ x_n.$$

Set

$$\beta =\bigcup _{n\ge 1}{\beta }_n$$

and let

$${\beta }^{*}$$

be its transitive closure.

Definition 11.

The equivalence relation ${\beta }^{*}$ is called thefundamental relation.

Under the standard hypergroup hypotheses, the quotient

$$P/{\beta }^{*}$$

is an ordinary group.

We now incorporate topology.

Let

$$q:P⟶P/{\beta }^{*}$$

be the quotient map, and equip $P/{\beta }^{*}$ with the quotient topology.

18. The Fundamental Quotient Theorem

Theorem 14

(Fundamental quotient). Let P be a Vietoris-complete multi-valued topological group. Suppose that every ${\beta }^{*}$-class is a complete part and that the quotient operation

$$\left[x\right]\left[y\right]=\left[z\right](z\in x\circ y)$$

is well defined. Then $P/{\beta }^{*}$ is a topological group with the quotient topology.

Proof. 

By the algebraic fundamental relation theorem,

$$P/{\beta }^{*}$$

is a group.

Let

$$q:P\to P/{\beta }^{*}$$

be the quotient map.

We first prove continuity of multiplication. Let

$$m\left(\right[x],[y\left]\right)=\left[z\right],z\in x\circ y.$$

Let W be an open neighborhood of $\left[z\right]$. Then

$$q^{-1}\left(W\right)$$

is open in P. Since the ${\beta }^{*}$-classes are complete parts, the quotient compatibility implies that multiplication of sufficiently small neighborhoods of x and y remains inside $q^{-1}\left(W\right)$.

By Vietoris continuity of ∘, there exist neighborhoods U of x and V of y such that

$$U\circ V\subseteq q^{-1}\left(W\right).$$

Consequently,

$$m\left(q\right(U),q(V\left)\right)\subseteq W.$$

Hence m is continuous.

For inversion, let

$$\overline{\iota }\left(\left[x\right]\right)=\left[x^{-1}\right].$$

Since inversion on P is continuous and q is a quotient map, the same argument gives continuity of $\overline{\iota }$.

Thus $P/{\beta }^{*}$ is a topological group. □

Remark 4.

The theorem shows that the multi-valued structure can be regarded as a topological enlargement of a genuine topological group. The fundamental quotient removes the ambiguity encoded by ${\beta }^{*}$.

19. Open Complete Parts and Quotient Maps

The following result is useful in applications.

Theorem 15.

Let P be Vietoris-complete and let H be a normal subpolygroup such that every open subset of P is a complete part. Then the natural map

$$q:P⟶P/H$$

is open.

Proof. 

Let U be open in P. Then

$$q^{-1}\left(q\left(U\right)\right)=H\circ U.$$

Since U is a complete part, the translation properties imply that $H\circ U$ is open. By the definition of the quotient topology, $q\left(U\right)$ is therefore open. □

Corollary 2.

Under the hypotheses of the theorem, $P/H$ is a quotient topological multi-valued group whenever the quotient hyperoperation is well defined.

20. Continuous Homomorphisms

Definition 12.

Let P and Q be multi-valued groups. A map

$$f:P\to Q$$

is astrong homomorphismif

$$f(x\circ y)=f\left(x\right)\circ f\left(y\right)$$

for every $x,y\in P$.

A weaker notion is often useful.

Definition 13.

A map $f:P\to Q$ is aweak homomorphismif

$$f(x\circ y)\subseteq f\left(x\right)\circ f\left(y\right)$$

for all $x,y\in P$.

Definition 14.

A strong homomorphism between Vietoris multi-valued topological groups is called a topological homomorphism if it is continuous.

Proposition 5.

The composition of continuous strong homomorphisms is a continuous strong homomorphism.

Proof. 

Let

$$f:P\to Q\mathrm{and}g:Q\to R$$

be continuous strong homomorphisms. Then

$$\begin{array}{cc}\hfill (g\circ f)(x\circ y)& =g\left(f\right(x\circ y\left)\right)\hfill \\ & =g\left(f\right(x)\circ f(y\left)\right)\hfill \\ & =g\left(f\right(x\left)\right)\circ g\left(f\right(y\left)\right)\hfill \\ & =(g\circ f)\left(x\right)\circ (g\circ f)\left(y\right).\hfill \end{array}$$

Continuity follows from continuity of f and g. □

21. First Isomorphism Theorem

Let

$$f:P\to Q$$

be a continuous strong homomorphism. Define

$$\ker \left(f\right)=\{x\in P:f\left(x\right)=e_Q\}.$$

Proposition 6.

The kernel of a continuous strong homomorphism is a closed subpolygroup whenever Q is Hausdorff.

Proof. 

Since

$$\left\{e_Q\right\}$$

is closed in Q and f is continuous,

$$\ker \left(f\right)=f^{-1}\left(\left\{e_Q\right\}\right)$$

is closed.

The algebraic properties follow from the homomorphism property and the polygroup axioms. □

Theorem 16

(First isomorphism theorem). Let

$$f:P\to Q$$

be a surjective continuous strong homomorphism between Hausdorff Vietoris multi-valued topological groups. Suppose that $\ker \left(f\right)$ is a normal subpolygroup and that the quotient operation is well defined. Then

$$P/\ker \left(f\right)$$

is topologically isomorphic to Q provided f is open.

Proof. 

The algebraic first isomorphism theorem gives an algebraic isomorphism

$$\overline{f}:P/\ker \left(f\right)\to Q,\overline{f}\left(\left[x\right]\right)=f\left(x\right).$$

It is well defined because elements in the same coset differ by an element of the kernel.

Since f is continuous and the quotient map is continuous, $\overline{f}$ is continuous. Since f is open, $\overline{f}$ is open. Hence $\overline{f}$ is a homeomorphism and an algebraic isomorphism. □

22. Metric Theory

Suppose now that $(P,d)$ is compact metric.

The Hausdorff metric on $\mathcal{K}\left(P\right)$ is

$$d_H(A,B)=\max \left\{\underset{a\in A}{\sup }d(a,B),\underset{b\in B}{\sup }d(b,A)\right\}.$$

Theorem 17.

Let $(P,d)$ be compact metric. A compact-valued operation

$$\circ :P\times P\to \mathcal{K}\left(P\right)$$

is Vietoris-continuous if and only if it is continuous with respect to the Hausdorff metric.

Proof. 

For compact metric spaces, the Hausdorff metric induces the Vietoris topology on the hyperspace of nonempty compact subsets. Hence the two notions of continuity coincide. □

Corollary 3.

Every compact metrizable Vietoris multi-valued topological group is uniformly multi-valued in the sense that for every $\epsilon >0$ there exists $\delta >0$ such that

$$d(x,x^{\prime })<\delta ,d(y,y^{\prime })<\delta$$

imply

$$d_H(x\circ y,x^{\prime }\circ y^{\prime })<\epsilon .$$

Proof. 

The product space $P\times P$ is compact. Therefore the continuous map

$$\circ :P\times P\to \mathcal{K}\left(P\right)$$

is uniformly continuous. □

23. A Compact Interval Example

Consider

$$P=[0,1].$$

Define

$$x\circ y=\left[|x-y|,\min \{1,x+y\}\right].$$

Proposition 7.

The operation above is Vietoris-continuous.

Proof. 

The endpoint maps

$$(x,y)\mapsto |x-y|$$

and

$$(x,y)\mapsto \min \{1,x+y\}$$

are continuous. Therefore the map into the space of compact intervals is continuous in the Hausdorff metric and hence in the Vietoris topology. □

Remark 5.

This example is not a polygroup. It is included to emphasize that topological regularity and algebraic group structure are logically independent requirements.

24. A Two-Valued Example

Let

$$P=\mathbb{T}=\{z\in \mathbb{C}:|z|=1\}$$

and let

$$\sigma \left(z\right)=\overline{z}.$$

Define

$$z\circ w=\{zw,\sigma \left(z\right)w\}=\{zw,\overline{z}w\}.$$

Proposition 8.

The operation

$$\circ :\mathbb{T}\times \mathbb{T}\to \mathcal{K}\left(\mathbb{T}\right)$$

is Vietoris-continuous.

Proof. 

The maps

$$(z,w)\mapsto zw$$

and

$$(z,w)\mapsto \overline{z}w$$

are continuous. A finite set-valued map obtained from finitely many continuous branches is Vietoris-continuous. □

Again, the operation need not satisfy the polygroup axioms. This example illustrates the distinction between a multi-valued topological magma and a multi-valued topological group.

25. Comparison with Convolution Hypergroups

A locally compact convolution hypergroup is fundamentally different from the present set-valued framework. Its product is described by probability measures:

$${\delta }_x*{\delta }_y.$$

The support

$$supp({\delta }_x*{\delta }_y)$$

does, however, produce a compact-valued set-valued operation in many standard examples.

Thus there is a natural forgetful correspondence

$${\delta }_x*{\delta }_y⟼supp({\delta }_x*{\delta }_y).$$

Proposition 9.

Suppose K is a compact convolution hypergroup. Then

$$x\circ y=supp({\delta }_x*{\delta }_y)$$

defines a compact-valued hyperoperation.

Proof. 

The convolution of two probability measures is a probability measure, and therefore has nonempty support. Since K is compact, the support is compact. □

Remark 6.

The support operation loses probabilistic information. Consequently, a convolution hypergroup contains strictly more information than the associated support-valued multi-valued structure.

26. Relationship with Existing Topological Hyperstructures

The present framework should be placed carefully relative to the existing literature.

Structure	Product	Topological feature
Topological group	$xy\in G$	multiplication and inversion continuous
Marty hypergroup	$x\circ y\subseteq G$	associative and reproductive
Polygroup	$x\circ y\subseteq G$	scalar identity and reversibility
Topological polygroup	$x\circ y\subseteq G$	hyperoperation and inverse continuous
Vietoris multi-valued topological group	$x\circ y\in \mathcal{K}\left(G\right)$	full Vietoris continuity and continuous inverse
Convolution hypergroup	${\delta }_x\ast {\delta }_y$	continuous probability-measure convolution

The topological polygroup theory is therefore the immediate algebraic predecessor of the present framework [6]. The principal additional restriction here is the compact-valued/full-Vietoris setting.

27. A Structural Decomposition Theorem

The fundamental quotient suggests a decomposition into an ordinary group component and an internal multi-valued component.

Let

$$q:P\to P/{\beta }^{*}$$

be the fundamental quotient.

For each $x\in P$, let

$${\left[x\right]}_{{\beta }^{*}}$$

denote its fundamental class.

Theorem 18.

Let P be a Vietoris-complete multi-valued topological group satisfying the hypotheses of the fundamental quotient theorem. Then every multi-valued product satisfies

$$x\circ y\subseteq q^{-1}\left(q\left(x\right)q\left(y\right)\right).$$

Moreover,

$$q(x\circ y)=\left\{q\right(x\left)q\right(y\left)\right\}.$$

Proof. 

If

$$z\in x\circ y,$$

then by the definition of the fundamental quotient,

$$\left[z\right]=\left[x\right]\left[y\right].$$

Hence

$$q\left(z\right)=q\left(x\right)q\left(y\right).$$

Therefore

$$x\circ y\subseteq q^{-1}\left(\left\{q\left(x\right)q\left(y\right)\right\}\right).$$

Since $x\circ y$ is nonempty, its image is precisely the singleton

$$\left\{q\right(x\left)q\right(y\left)\right\}.$$

□

Remark 7.

The theorem shows that the quotient removes precisely the multi-valuedness visible to the fundamental relation. The remaining multi-valued behavior is therefore internal to the fundamental classes.

28. A Criterion for Complete Fibers

Proposition 10.

Suppose that every fiber of the fundamental quotient

$$q:P\to P/{\beta }^{*}$$

is a complete part. If $U\subseteq P/{\beta }^{*}$ is open, then

$$q^{-1}\left(U\right)$$

is a complete part of P.

Proof. 

Let

$$x_1,\dots ,x_n\in P$$

and suppose

$$q^{-1}\left(U\right)\cap (x_1\circ \dots \circ x_n)\ne ⌀.$$

Choose

$$z\in q^{-1}\left(U\right)\cap (x_1\circ \dots \circ x_n).$$

Then

$$q\left(z\right)\in U.$$

Every element of

$$x_1\circ \dots \circ x_n$$

belongs to the same fundamental class as z. Hence its image under q equals $q\left(z\right)\in U$. Therefore

$$x_1\circ \dots \circ x_n\subseteq q^{-1}\left(U\right).$$

Thus $q^{-1}\left(U\right)$ is complete. □

29. Compact Generation

Definition 15.

A Vietoris multi-valued topological group P is compactly generated if there exists a family

$$\{K_i:i\in I\}$$

of compact subpolygroups such that

$$P=\bigcup _{i\in I}{K}_i.$$

Proposition 11.

Every compact Vietoris multi-valued topological group is compactly generated.

Proof. 

Take the one-element family

$$K_1=P.$$

□

The concept becomes interesting for noncompact spaces.

Question 19.

Under what conditions does a locally compact Vietoris multi-valued topological group admit a compact exhaustion by subpolygroups?

This question is an analogue of compact-generation problems in topological group theory.

30. Completion

The uniformities introduced earlier suggest a completion theory.

Let

$$(P,{\mathcal{U}}_L)$$

be the left uniform space of a Vietoris-complete multi-valued topological group.

A left Cauchy filter $\mathcal{F}$ is defined in the usual uniform sense: for every entourage E there exists $A\in \mathcal{F}$ such that

$$A\times A\subseteq E.$$

Question 20.

Does every Hausdorff Vietoris-complete multi-valued topological group admit a completion $\widehat{P}$ such that:

1.

P embeds densely into $\widehat{P}$;

2.

the multi-valued operation extends continuously to $\widehat{P}\times \widehat{P}$;

3.

the extension is compact-valued;

4.

$\widehat{P}$ is complete with respect to the canonical uniformity?

For ordinary topological groups this is closely related to the theory of group completions. The multi-valued case appears to require additional control of compact-valued limits.

31. Invariant Measures

For a finite-valued multi-valued group, one can associate averaging operators. Suppose

$$x\circ y=\{z_1,\dots ,z_n\}.$$

Define

$$T_{x}f\left(y\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n}f\left(z_j\right).$$

More generally, one may introduce weights

$$p_j(x,y)\ge 0,{\displaystyle \sum _{j=1}^n}{p}_j(x,y)=1,$$

and define

$$T_{x}f\left(y\right)={\displaystyle \sum _{j=1}^n}{p}_j(x,y)f\left(z_j\right).$$

This suggests a measure-valued enrichment of the theory.

One can ask when does a locally compact Vietoris multi-valued topological group admit a probability kernel $K(x,y,·)$ whose support is exactly $x\circ y$ and which is associative in the convolution sense?

In the next section we try to discuss this since a positive answer would provide a bridge from the set-valued theory to Dunkl–Jewett–Spector hypergroups [4,7,14].

32. Probability Kernels Compatible with Vietoris Multi-Valued Topological Groups

Let G be a locally compact Hausdorff space, and let

$$\circ :G\times G⟶\mathcal{K}\left(G\right)$$

be a set-valued multiplication, where $\mathcal{K}\left(G\right)$ denotes the family of nonempty compact subsets of G, endowed with the Vietoris topology. We investigate the existence of a probability kernel

$$K:G\times G\times \mathcal{B}\left(G\right)⟶[0,1],(x,y,A)⟼K(x,y,A),$$

such that

$$suppK(x,y,·)=x\circ y$$

for all $x,y\in G$, and such that the induced convolution is associative.

Here $\mathcal{B}\left(G\right)$ denotes the Borel $\sigma$-algebra of G. The support of a probability measure is the smallest closed set of full measure.

32.1. Probability Kernels and Convolution

For Borel probability measures $\mu$ and $\nu$ on G, define their convolution by

$$\left(\mu {\ast }_K\nu \right)\left(A\right)={\int }_G{\int }_{G}K(x,y,A)d\mu \left(x\right)d\nu \left(y\right),A\in \mathcal{B}\left(G\right).$$

(1)

Equivalently, for a bounded Borel function $f:G\to \mathbb{R}$, put

$$Kf(x,y)={\int }_{G}f\left(z\right)K(x,y,dz).$$

Then

$${\int }_{G}fd\left(\mu {\ast }_K\nu \right)={\int }_G{\int }_{G}Kf(x,y)d\mu \left(x\right)d\nu \left(y\right).$$

The fundamental associativity condition is

$${\int }_{G}K(u,w,A)K(x,y,du)={\int }_{G}K(x,v,A)K(y,w,dv)$$

(2)

for all $x,y,w\in G$ and all $A\in \mathcal{B}\left(G\right)$. The left-hand side corresponds to first multiplying x and y, and then multiplying the result by w. The right-hand side corresponds to first multiplying y and w, and then multiplying x by the result.

Under the usual measurability hypotheses, this identity implies

$$\left(\mu {\ast }_K\nu \right){\ast }_K\lambda =\mu {\ast }_K\left(\nu {\ast }_K\lambda \right)$$

for all Borel probability measures $\mu ,\nu ,\lambda$.

32.2. A Sufficient-Existence Theorem

Theorem 21.

Let G be a locally compact Hausdorff second-countable space, and let

$$\circ :G\times G⟶\mathcal{K}\left(G\right)$$

be Vietoris-continuous. Suppose there exists a map

$$P:G\times G⟶\mathcal{P}\left(G\right),(x,y)⟼P_{x,y},$$

satisfying the following conditions:

(i)

For every $x,y\in G$, $P_{x,y}$ is a Borel probability measure and

$$supp{P}_{x,y}=x\circ y.$$

(ii)

The map $(x,y)\mapsto P_{x,y}$ is continuous for the weak topology on probability measures. Equivalently, for every $f\in C_b\left(G\right)$, the function

$$(x,y)⟼{\int }_{G}f\left(z\right)P_{x,y}\left(dz\right)$$

is continuous.

(iii)

For all $x,y,w\in G$ and all $f\in C_b\left(G\right)$,

$$\begin{array}{cc}\hfill & {\int }_G\left({\int }_{G}f\left(t\right)P_{u,w}\left(dt\right)\right)P_{x,y}\left(du\right)\hfill \\ \hfill & ={\int }_G\left({\int }_{G}f\left(t\right)P_{x,v}\left(dt\right)\right)P_{y,w}\left(dv\right).\hfill \end{array}$$

(3)

Then

$$K(x,y,A):=P_{x,y}\left(A\right)$$

defines a weakly continuous probability kernel whose support is exactly $x\circ y$. Moreover, its induced convolution is associative on the space of Borel probability measures on G.

Proof. 

By condition (i), $P_{x,y}$ is a probability measure supported exactly on $x\circ y$. Since G is a second-countable locally compact Hausdorff space, it is a standard Borel space. The weakly continuous map $(x,y)\mapsto P_{x,y}$ is Borel measurable as a map into the space of probability measures. Consequently, $(x,y)\mapsto P_{x,y}\left(A\right)$ is Borel measurable for every $A\in \mathcal{B}\left(G\right)$, so K is a probability kernel.

Condition (ii) gives the asserted weak continuity. Condition (iii) states that the two probability measures obtained by the two possible successive kernel operations agree when integrated against every bounded continuous function. Since bounded continuous functions determine Borel probability measures on a metrizable space, the two measures coincide. Thus the pointwise associativity identity holds.

Finally, applying Tonelli’s theorem to the iterated integrals in the definition of convolution gives

$$\left(\mu {\ast }_K\nu \right){\ast }_K\lambda =\mu {\ast }_K\left(\nu {\ast }_K\lambda \right).$$

Hence the convolution is associative. □

Remark 8.

The theorem gives a sufficient criterion, rather than an automatic existence result. Its essential hypothesis is the existence of a weakly continuous family of full-support probability measures satisfying the associativity identity. Local compactness and Vietoris continuity alone do not establish this compatibility.

32.3. Constructing Measures with the Required Support

The support requirement can be satisfied pointwise under relatively mild hypotheses.

Proposition 12.

Let G be a locally compact, second-countable Hausdorff space. For every nonempty compact subset $F\subseteq G$, there exists a Borel probability measure ${\nu }_F$ such that

$$supp{\nu }_F=F.$$

Proof. 

Since F is compact and metrizable, it is separable. Choose a countable dense sequence ${\left(a_n\right)}_{n\ge 1}$ in F and define

$${\nu }_F={\displaystyle \sum _{n=1}^{\infty }}{2}^{-n}{\delta }_{a_n},$$

where ${\delta }_{a_n}$ denotes the Dirac probability measure at $a_n$. Because ${\sum }_{n=1}^{\infty }{2}^{-n}=1$, the measure ${\nu }_F$ is a probability measure.

Every open neighborhood in G of a point of F intersects F in a nonempty relatively open set. By density, this set contains some $a_n$, and hence the neighborhood has positive ${\nu }_F$-measure. Therefore every point of F belongs to $supp{\nu }_F$. Since ${\nu }_F\left(F\right)=1$ and F is closed, its support is contained in F. Consequently,

$$supp{\nu }_F=F.$$

□

Applying Proposition 12 to each compact set $x\circ y$ yields a probability measure with the desired support for every pair $(x,y)$. However, this pointwise construction does not guarantee that the measures can be chosen weakly continuously in $(x,y)$. Nor does it guarantee the associativity identity. These are separate requirements.

32.4. The Deterministic Special Case

A simple sufficient condition is obtained when the multi-valued operation reduces to an ordinary continuous associative operation.

Corollary 4.

Suppose $m:G\times G\to G$ is continuous and associative, and

$$x\circ y=\left\{m\right(x,y\left)\right\}$$

for all $x,y\in G$. Then

$$K(x,y,·)={\delta }_{m(x,y)}$$

is a weakly continuous, associative probability kernel satisfying

$$suppK(x,y,·)=x\circ y.$$

Proof. 

The Dirac measure ${\delta }_{m(x,y)}$ has support $\left\{m\right(x,y\left)\right\}=x\circ y$. Continuity of m implies weak continuity of the kernel. Moreover, for every bounded continuous $f:G\to \mathbb{R}$, the two sides of the associativity identity reduce to

$$f\left(m\right(m(x,y),w\left)\right)\mathrm{and}f\left(m\right(x,m(y,w)\left)\right),$$

respectively. These expressions are equal because m is associative. The induced convolution is therefore associative. □

A locally compact Vietoris multi-valued topological group admits the requested probability kernel whenever a family of probability measures can be constructed that satisfies three requirements:

$$\begin{array}{c}suppK(x,y,·)=x\circ y,\\ (x,y)⟼K(x,y,·)\mathrm{is}\mathrm{weakly}\mathrm{continuous},\\ {\int }_{G}K(u,w,·)K(x,y,du)={\int }_{G}K(x,v,·)K(y,w,dv).\end{array}$$

The last equality is understood as equality of probability measures. The first requirement concerns the support of each individual measure, the second concerns topological regularity, and the third ensures associativity of convolution. All three must be verified; none follows from Vietoris continuity alone.

33. Conclusions

We have proposed a strengthened topological framework for group-like multi-valued algebra. The central object is a polygroup equipped with a compact-valued hyperoperation

$$\circ :P\times P⟶\mathcal{K}\left(P\right)$$

which is continuous in the full Vietoris topology.

This formulation has several advantages. First, it separates the algebraic issue of multi-valued multiplication from the topological issue of hyperspace continuity. Second, compact-valuedness makes the Vietoris topology particularly natural and permits closed-graph and compactness results. Third, the use of polygroup axioms gives a genuine group-like inverse structure and permits the construction of fundamental quotients.

The complete-part condition provides an important bridge between local topological structure and algebraic structure. Under this hypothesis, translations generate neighborhood bases and canonical left and right uniformities can be constructed. The fundamental relation then provides a mechanism by which the genuinely multi-valued structure can be reduced to an ordinary topological group.

The singleton-collapse theorem shows that ordinary topological groups are recovered exactly when the multi-valued product degenerates to singletons.

The resulting picture can therefore be summarized schematically as

$$\begin{array}{|c|}\hline \begin{array}{c}\mathrm{ordinary}\mathrm{topological}\mathrm{groups}\\ \mathrm{U}\\ \mathrm{Vietoris}\mathrm{multi}-\mathrm{valued}\mathrm{topological}\mathrm{groups}\\ \mathrm{U}\\ \mathrm{topological}\mathrm{hyperstructures}\\ \mathrm{U}\\ \mathrm{general}\mathrm{set}-\mathrm{valued}\mathrm{algebraic}\mathrm{systems}.\end{array}\\ \hline\end{array}$$

At the same time, the support map

$${\delta }_x\ast {\delta }_y⟼supp({\delta }_x\ast {\delta }_y)$$

connects the present set-valued theory with the analytic theory of convolution hypergroups.