Boundary-Approach Density of Compactification Remainders, Products, Order and Spectra

1. Introduction

Let X be a Tychonoff space and $bX$ a Hausdorff compactification. The theory of remainders studies the subspace $bX\setminus X$ and how its topology reflects properties of X. For general background, see [1,2,3,4,5]. We write $d_b^{\partial }\left(X\right)$ for an embedding-sensitive cardinal that records how the dense subspace approaches the entire remainder inside the chosen compactification. Its formal definition appears in Definition 2.1. The defining requirement is that a single witnessing set accumulate at every remainder point. Closely related earlier notions include Tkachenko’s relative density ([6], p. 43), the domination terminology of Gruenhage, Tkachuk and Wilson ([7], Section 1), and the selection approaching number introduced by García-Ferreira et al. ([8], Section 4) and later called the approaching number by Gutev ([9], p. 89). The distinction is made precise immediately after Definition 2.1.

Boundary-approach density also differs from the ordinary density of the ambient compactification. Van Douwen showed that all compactifications of a fixed space have the same ordinary density [10]. Another compactification-based cardinal, the source number, is likewise independent of the chosen compactification ([11], Proposition 2.2). Boundary-approach density, in contrast, may have a full cardinal spectrum. The double density spectrum of a space consists of the densities of its dense subspaces ([12], p. 384), whereas the boundary-approach spectrum arises as the compactification of X varies.

The first main theorem gives an exact product trichotomy. Products of compactifications and their remainders have a substantial literature [13,14,15,16,17]. For comparison, the selection approaching number and the source number have different product behavior. For the selection approaching number $sa$, García-Ferreira et al. ([8], Section 4) observed the pointwise inequality

$$sa\left(\right(p,q),X\times Y)\le max\left\{sa\right(p,X),sa(q,Y\left)\right\},$$

whereas for the source number $so$, Wang and Zhang ([11], Proposition 2.10) proved that

$$so\left(\prod _{\alpha <\kappa }{X}_{\alpha }\right)\le \kappa ⟺so\left(X_{\alpha }\right)\le \kappa (\alpha <\kappa )$$

for every infinite cardinal $\kappa$. Neither result computes a cardinal attached to a specified factorwise compactification, nor does either require a single subset of the product to accumulate at every point of the remainder. For a non-empty family ${\left(X_i\right)}_{i\in I}$ of non-empty Tychonoff spaces with Hausdorff compactifications $b_i{X}_i$, put $P={\prod }_{i\in I}{X}_i$ and $K={\prod }_{i\in I}{b}_i{X}_i$. Theorem 5.1 gives a complete trichotomy by the number of non-compact factors. If at least two factors are non-compact, then

$$d_K^{\partial }\left(P\right)=dens\left(P\right),$$

independently of the chosen factor compactifications. With exactly one non-compact factor $X_j$, put $P_{-j}={\prod }_{i\in I\setminus \left\{j\right\}}{X}_i$. Then the value is

$$max\{d_{b_j}^{\partial }\left(X_j\right),dens\left(P_{-j}\right)\}.$$

This rigidity has a spectral consequence for finite products of locally compact metrizable spaces with at least two non-compact factors. Factorwise product compactifications realize the maximum of the spectrum. If $dens\left(P\right)>\omega$, every smaller infinite value is realized, but only by non-factorwise compactifications.

The second main result concerns the boundary-approach spectrum ${\Sigma }_{\partial }\left(X\right)$, formally defined in Definition 2.2. For a non-compact locally compact Tychonoff space X, the one-point compactification $\alpha X$ and the Stone–Čech compactification $\beta X$ attain the minimum and maximum. The lower endpoint is

$$d_{\alpha }^{\partial }\left(X\right)=min\left\{\right|D|:D\subset X\mathrm{and}{cl}_{X}D\mathrm{is}\mathrm{non}-\mathrm{compact}\},$$

and hence $d_{\alpha }^{\partial }\left(X\right)=\omega$ exactly when X is not $\omega$-bounded. The standard notion of $\omega$-boundedness is recalled in Section 2. Theorem 6.3 gives a general construction realizing an interval of values. If a non-compact locally compact Tychonoff space X has a clopen partition

$$X=\coprod _{\xi <\tau }{G}_{\xi },$$

where $\tau$ is infinite and each $G_{\xi }$ is non-empty and non-$\omega$-bounded, then the infinite cardinals $\lambda \le \tau$ are simultaneously realized by compactifications $b_{\lambda }X$ satisfying $b_{\lambda }X\le b_{\mu }X$ whenever $\omega \le \lambda \le \mu \le \tau$. If $\tau =dens\left(X\right)$, the boundary-approach spectrum is the full interval from $\omega$ to $dens\left(X\right)$. When $dens\left(X\right)>\omega$, the chain may be chosen with endpoints $\alpha X$ and $\beta X$. In particular, the same full-spectrum conclusion holds for every non-compact locally compact metrizable space. Classical graph-closure constructions for prescribed remainders provide background for the compactification side of the argument [18]. Compactifications over discrete spaces have also been studied [19]. Theorem 6.3 simultaneously realizes every infinite boundary-approach value in the stated interval and arranges the realizing compactifications in an order-coherent chain.

Theorem 2.5 converts the ambient closure condition into a closed-core condition using functions that extend over $bX$. For $bX=\beta X$, the criterion becomes intrinsic and can be expressed in terms of complete separation. Such criteria are classical in compactification theory. Hewitt characterized spaces with a unique compactification by the fact that two completely separated closed sets cannot both be non-compact [20]. The present formula is quantitative and relative to a fixed compactification. In particular, it gives

$$d_{\beta }^{\partial }\left(X\right)=dens\left(X\right)$$

for every non-compact metrizable X. Boundary-approach density is also monotone in the compactification order, and a related inequality holds under an extension criterion for maps. For Stone–Čech compactifications, every continuous map $f:X\to Y$ with dense image therefore satisfies

$$d_{\beta }^{\partial }\left(Y\right)\le d_{\beta }^{\partial }\left(X\right).$$

Boundary-approach density can exceed the ordinary density of the ambient compactification. Using an example of Levy and McDowell ([21], p. 426), Corollary 5.3 yields a compactification K of a space P for which

$$dens\left(K\right)=\omega <d_K^{\partial }\left(P\right).$$

This strict gap separates boundary-approach density from the ordinary density of the ambient compactification.

Throughout, ⊂ denotes inclusion, not necessarily proper. All cardinal minima in this paper are taken literally, so finite values are not replaced by $\omega$.

2. Preliminaries and Closed Cores

We identify X with its canonical dense copy in $bX$ and write

$$B_b\left(X\right)=bX\setminus X.$$

Definition 2.1. 

For a Hausdorff compactification $bX$ of a Tychonoff space X, define

$$d_b^{\partial }\left(X\right)=min\left\{\right|D|:D\subset X\mathrm{and}{B}_b\left(X\right)\subset {cl}_{bX}D\}.$$

If $B_b\left(X\right)=⌀$, put $d_b^{\partial }\left(X\right)=0$.

Definition 2.2. 

For a Tychonoff space X, define itsboundary-approach spectrumby

$${\Sigma }_{\partial }\left(X\right)=\{d_c^{\partial }\left(X\right):cX\mathrm{is}\mathrm{a}\mathrm{Hausdorff}\mathrm{compactification}\mathrm{of}X\}.$$

The closure condition in Definition 2.1 is closely related to several earlier notions. Tkachenko defined the density of a subspace Y in an ambient space Z as the least size of a set $M\subset Z$ with $Y\subset {cl}_{Z}M$ ([6], p. 43). In modern terminology, Gruenhage, Tkachuk and Wilson say that AdominatesB when $B\subset clA$ ([7], Section 1). Here the target is $B_b\left(X\right)$, while every witnessing set is required to lie in the specified dense subspace X. This restriction to witnesses inside X is essential. If $B_b\left(X\right)=\left\{p\right\}$, Tkachenko’s quantity for $\left\{p\right\}\subset bX$ equals 1 (take $M=\left\{p\right\}$), whereas $d_b^{\partial }\left(X\right)$ is the approaching number of p in $bX$ and is infinite. This pointwise invariant was introduced under the name selection approaching number by García-Ferreira et al. ([8], Section 4) and later called the approaching number by Gutev ([9], p. 89).

The minimum exists because X is dense in $bX$. If D is dense in X, then it is dense in $bX$, and hence

$$d_b^{\partial }\left(X\right)\le dens\left(X\right).$$

(1)

If X is non-compact, then $B_b\left(X\right)\ne ⌀$ and $d_b^{\partial }\left(X\right)$ is infinite, since finite subsets of the Hausdorff space $bX$ are closed.

Recall that X is $\omega$-bounded if every countable subset of X has compact closure [22]. For compactifications $bX$ and $cX$ of the same space, write $bX\le cX$ if there is a continuous map $\pi :cX\to bX$ whose restriction to X is the identity. This is the standard compactification order, whose largest element is $\beta X$ [3,23].

Definition 2.3. 

For $A,F\subset X$, say that A and F areb-separatedif there is a continuous function $u:bX\to [0,1]$ such that $u\left[A\right]\subset \left\{0\right\}$ and $u\left[F\right]\subset \left\{1\right\}$.

Lemma 2.4. 

Let $A\subset X$. Then

$$B_b\left(X\right)\subset {cl}_{bX}A$$

if and only if every closed subset $F\subset X$ that is b-separated from A is compact.

Proof. 

Suppose first that $B_b\left(X\right)\subset {cl}_{bX}A$, and let $F\subset X$ be closed and b-separated from A. Choose $u:bX\to [0,1]$ with $u\left[A\right]\subset \left\{0\right\}$ and $u\left[F\right]\subset \left\{1\right\}$. Then

$${cl}_{bX}A\subset u^{-1}\left(0\right),{cl}_{bX}F\subset u^{-1}\left(1\right),$$

so the two closures are disjoint. Hence ${cl}_{bX}F$ misses $B_b\left(X\right)$ and is contained in X. Since F is closed in X, it follows that $F={cl}_{bX}F$, and therefore F is compact.

Conversely, suppose $p\in B_b\left(X\right)\setminus {cl}_{bX}A$. Compact Hausdorff spaces are normal, so there is a continuous $u:bX\to [0,1]$ such that

$$u\left[{cl}_{bX}A\right]\subset \left\{0\right\},u\left(p\right)=1.$$

Put

$$F=X\cap u^{-1}\left([1/2,1]\right).$$

Then F is closed in X and is also b-separated from A, since composing u with a continuous map of $[0,1]$ that sends 0 to 0 and is identically 1 on $[1/2,1]$ gives a witnessing function. Moreover, $p\in {cl}_{bX}F$ because every neighborhood of p contains a smaller neighborhood on which $u>1/2$, and that smaller neighborhood meets the dense set X. Thus F is not compact. If it were compact, it would be closed in the Hausdorff space $bX$, which is impossible because its closure contains $p\notin X$. □

Theorem 2.5. 

For every Tychonoff space X and every Hausdorff compactification $bX$,

$$d_b^{\partial }\left(X\right)=min\left\{dens\left(A\right):\begin{array}{c}A\subset X\mathrm{is}\mathrm{closed},\mathrm{and}\hfill \\ \mathrm{every}\mathrm{closed}F\subset X\mathrm{that}\mathrm{is}b-\mathrm{separated}\hfill \\ \mathrm{from}A\mathrm{is}\mathrm{compact}\hfill \end{array}\right\}.$$

Here $dens(⌀)=0$.

Proof. 

Let $D\subset X$ witness $d_b^{\partial }\left(X\right)$ and put $A={cl}_{X}D$. Then A is closed, $dens\left(A\right)\le \left|D\right|$, and

$$B_b\left(X\right)\subset {cl}_{bX}D\subset {cl}_{bX}A.$$

Lemma 2.4 shows that A has the stated property. This proves one inequality.

Conversely, let A be closed and satisfy the stated property. Choose $D\subset A$ dense in A with $\left|D\right|=dens\left(A\right)$. Lemma 2.4 gives $B_b\left(X\right)\subset {cl}_{bX}A$. Since $A\subset {cl}_{bX}D$ and ${cl}_{bX}D$ is closed,

$$B_b\left(X\right)\subset {cl}_{bX}A\subset {cl}_{bX}D.$$

Thus D is a boundary-approach witness. Taking minima proves the formula. □

3. The Stone–Čech Case

Two subsets of a Tychonoff space are completely separated if a continuous function $X\to [0,1]$ is 0 on one and 1 on the other. Every such function extends continuously over $\beta X$ [2,3,23]. Theorem 2.5 therefore takes the following intrinsic form in the Stone–Čech case.

Corollary 3.1. 

For every Tychonoff space X,

$$d_{\beta }^{\partial }\left(X\right)=min\left\{dens\left(A\right):\begin{array}{c}A\subset X\mathrm{is}\mathrm{closed},\mathrm{and}\hfill \\ \mathrm{every}\mathrm{closed}F\subset X\mathrm{completely}\mathrm{separated}\hfill \\ \mathrm{from}A\mathrm{is}\mathrm{compact}\hfill \end{array}\right\}.$$

Consequently, $d_{\beta }^{\partial }\left(X\right)\le \omega$ if and only if X contains a closed separable set A such that every closed set completely separated from A is compact.

Corollary 3.2. 

If X is normal and Tychonoff, then

$$d_{\beta }^{\partial }\left(X\right)=min\left\{dens\left(A\right):\begin{array}{c}A\subset X\mathrm{is}\mathrm{closed},\mathrm{and}\mathrm{every}\mathrm{closed}F\subset X\hfill \\ \mathrm{disjoint}\mathrm{from}A\mathrm{is}\mathrm{compact}\hfill \end{array}\right\}.$$

Proof. 

In a normal space, any two disjoint closed sets are completely separated by Urysohn’s lemma. Apply Corollary 3.1. □

Proposition 3.3. 

If X is a non-compact metrizable space, then

$$d_{\beta }^{\partial }\left(X\right)=dens\left(X\right).$$

Hence $d_{\beta }^{\partial }\left(X\right)\le \omega$ if and only if X is separable.

Proof. 

The inequality $d_{\beta }^{\partial }\left(X\right)\le dens\left(X\right)$ follows from (1). For the reverse inequality, choose a non-empty closed set $A\subset X$ that satisfies the condition in Corollary 3.2 and has

$$dens\left(A\right)=d_{\beta }^{\partial }\left(X\right).$$

Fix a compatible metric d on X and, for $n\ge 1$, put

$$F_n=\{x\in X:d(x,A)\ge 1/n\}.$$

Each $F_n$ is closed and disjoint from A, hence compact. Compact metrizable spaces are separable, and

$$X=A\cup {\displaystyle \bigcup _{n=1}^{\infty }}{F}_n.$$

Therefore

$$dens\left(X\right)\le max\{dens(A),\omega \}.$$

Since X is non-compact, $d_{\beta }^{\partial }\left(X\right)$ is infinite, and the right-hand side equals $d_{\beta }^{\partial }\left(X\right)$. Combining this with (1) gives equality. □

4. Compactification Order and Maps

We begin with monotonicity for the compactification order defined in Section 2.

Proposition 4.1. 

If $bX\le cX$, then

$$d_b^{\partial }\left(X\right)\le d_c^{\partial }\left(X\right).$$

In particular, $d_b^{\partial }\left(X\right)\le d_{\beta }^{\partial }\left(X\right)$ for every Hausdorff compactification $bX$.

Proof. 

Let $\pi :cX\to bX$ fix X pointwise. Its image is compact and contains the dense set X, so $\pi$ is onto. We first show

$$\pi [cX\setminus X]=bX\setminus X.$$

Any preimage of a point in $bX\setminus X$ lies outside X. Conversely, suppose $p\in cX\setminus X$ and $\pi \left(p\right)=x\in X$. Choose disjoint open neighborhoods $U_p,U_x$ of $p,x$ in $cX$. Since $U_x\cap X$ is a neighborhood of x in X, choose an open neighborhood V of x in $bX$ such that

$$V\cap X\subset U_x\cap X.$$

Then $U_p\cap {\pi }^{-1}\left(V\right)$ is a neighborhood of p and hence meets the dense set X. Choose $y\in U_p\cap {\pi }^{-1}\left(V\right)\cap X$. Then

$$y=\pi \left(y\right)\in V\cap X\subset U_x,$$

contradicting $y\in U_p$.

Now let $D\subset X$ satisfy $cX\setminus X\subset {cl}_{cX}D$. Continuity gives

$$bX\setminus X=\pi [cX\setminus X]\subset \pi \left[{cl}_{cX}D\right]\subset {cl}_{bX}D.$$

Thus every witness for $cX$ is a witness for $bX$. □

Proposition 4.2. 

Let $bX$ and $cY$ be compactifications of Tychonoff spaces X and Y. Suppose that $f:X\to Y$ has a continuous extension $\overline{f}:bX\to cY$ satisfying

$$cY\setminus Y\subset \overline{f}[bX\setminus X].$$

Then

$$d_c^{\partial }\left(Y\right)\le d_b^{\partial }\left(X\right).$$

Proof. 

If $bX\setminus X\subset {cl}_{bX}D$, then

$$cY\setminus Y\subset \overline{f}[bX\setminus X]\subset \overline{f}\left[{cl}_{bX}D\right]\subset {cl}_{cY}f\left[D\right].$$

Hence $f\left[D\right]$ is a witness and $\left|f\right[D\left]\right|\le \left|D\right|$. □

Corollary 4.3. 

Let $f:X\to Y$ be a continuous map between Tychonoff spaces. If $f\left[X\right]$ is dense in Y, then

$$d_{\beta }^{\partial }\left(Y\right)\le d_{\beta }^{\partial }\left(X\right).$$

In particular, the inequality holds for every continuous surjection.

Proof. 

Let $\beta f:\beta X\to \beta Y$ be the Stone–Čech extension [3]. Its image is compact and therefore closed in $\beta Y$. It contains $f\left[X\right]$, which is dense in Y and hence dense in $\beta Y$. Thus $\beta f$ is onto. If $q\in \beta Y\setminus Y$ and $\beta f\left(p\right)=q$, then $p\notin X$, because $f\left[X\right]\subset Y$. Therefore

$$\beta Y\setminus Y\subset \beta f[\beta X\setminus X].$$

Apply Proposition 4.2. □

5. Products

For factorwise product compactifications, the behavior is particularly rigid. For an index set I and $j\in I$, write

$$P_{-j}=\prod _{i\in I\setminus \left\{j\right\}}{X}_i,$$

with the empty product interpreted as a singleton.

Theorem 5.1. 

Let ${\left(X_i\right)}_{i\in I}$ be a non-empty family of non-empty Tychonoff spaces, let $b_i{X}_i$ be a Hausdorff compactification of $X_i$, and put

$$P=\prod _{i\in I}{X}_i,K=\prod _{i\in I}{b}_i{X}_i.$$

Let

$$N=\{i\in I:X_i\mathrm{is}\mathrm{non}-\mathrm{compact}\}.$$

The following cases exhaust all possibilities.

(i)

If $N=⌀$, then $P=K$ and $d_K^{\partial }\left(P\right)=0$.

(ii)

If $N=\left\{j\right\}$, then

$$d_K^{\partial }\left(P\right)=max\{d_{b_j}^{\partial }\left(X_j\right),dens\left(P_{-j}\right)\}.$$

(iii)

If $\left|N\right|\ge 2$, then

$$d_K^{\partial }\left(P\right)=dens\left(P\right).$$

Proof. 

Part (i) is immediate. Suppose $N=\left\{j\right\}$. Then every $X_i$ with $i\ne j$ is compact, hence $b_i{X}_i=X_i$, and

$$K\setminus P=(b_j{X}_j\setminus X_j)\times P_{-j}.$$

Let $H\subset P$ satisfy $K\setminus P\subset {cl}_{K}H$. Projecting onto the jth coordinate gives

$$\left|H\right|\ge d_{b_j}^{\partial }\left(X_j\right).$$

Let ${\pi }_{-j}:K\to P_{-j}$ be the projection onto the coordinates other than j. Fix $p\in b_j{X}_j\setminus X_j$. The slice $\left\{p\right\}\times P_{-j}$ lies in $K\setminus P$, so projecting away from the jth coordinate gives

$$P_{-j}\subset {cl}_{P_{-j}}{\pi }_{-j}\left[H\right].$$

Thus $\left|H\right|\ge dens\left(P_{-j}\right)$.

Conversely, take a boundary-approach witness $D_j\subset X_j$ of size $d_{b_j}^{\partial }\left(X_j\right)$ and a dense set $E\subset P_{-j}$ of size $dens\left(P_{-j}\right)$. Then

$$(b_j{X}_j\setminus X_j)\times P_{-j}\subset {cl}_K(D_j\times E).$$

Since $X_j$ is non-compact, $d_{b_j}^{\partial }\left(X_j\right)$ is infinite, and therefore

$$|D_j\times E|=max\{d_{b_j}^{\partial }\left(X_j\right),dens\left(P_{-j}\right)\}.$$

This proves (ii).

Suppose now that $\left|N\right|\ge 2$, and let $H\subset P$ be a boundary-approach witness. Choose distinct $j,k\in N$ and points

$$p_j\in b_j{X}_j\setminus X_j,p_k\in b_k{X}_k\setminus X_k.$$

For $r\in \{j,k\}$, let $K_{-r}={\prod }_{i\in I\setminus \left\{r\right\}}{b}_i{X}_i$ and let ${\rho }_{-r}:K\to K_{-r}$ be the coordinate projection. Since $\left\{p_r\right\}\times P_{-r}\subset K\setminus P\subset {cl}_{K}H$, continuity and the inclusion ${\rho }_{-r}\left[H\right]\subset P_{-r}$ give $P_{-r}\subset {cl}_{K_{-r}}{\rho }_{-r}\left[H\right]\cap P_{-r}={cl}_{P_{-r}}{\rho }_{-r}\left[H\right]$, and hence $\left|H\right|\ge dens\left(P_{-r}\right)$ for $r=j,k$. The coordinate projection from $P_{-k}$ onto $X_j$ is surjective, so $dens\left(X_j\right)\le dens\left(P_{-k}\right)$. Because $X_j$ is a non-compact Tychonoff space, $dens\left(X_j\right)$ is infinite. The space $P_{-j}$ projects onto the non-compact factor $X_k$, so $dens\left(P_{-j}\right)$ is infinite as well. Hence

$$\begin{array}{cc}\hfill dens\left(P\right)& \le dens\left(X_j\right)·dens\left(P_{-j}\right)\hfill \\ & =max\{dens\left(X_j\right),dens\left(P_{-j}\right)\}\hfill \\ & \le max\{dens\left(P_{-k}\right),dens\left(P_{-j}\right)\}\le \left|H\right|.\hfill \end{array}$$

Because P is dense in K, every dense subset of P is dense in K. Hence $d_K^{\partial }\left(P\right)\le dens\left(P\right)$, proving (iii). □

Corollary 5.2. 

If X and Y are non-compact Tychonoff spaces and $bX,cY$ are arbitrary Hausdorff compactifications, then

$$d_{bX\times cY}^{\partial }(X\times Y)=dens(X\times Y)=max\{dens\left(X\right),dens\left(Y\right)\}.$$

In particular, for product compactifications of two non-compact factors, the value is independent of the chosen compactifications.

Corollary 5.3. 

There exist a Tychonoff space P and a Hausdorff compactification K of P such that

$$dens\left(K\right)=\omega <d_K^{\partial }\left(P\right).$$

Proof. 

Levy and McDowell constructed a non-separable Tychonoff space Z whose Stone–Čech compactification $\beta Z$ is separable ([21], p. 426). Thus Z is non-compact, since otherwise $\beta Z=Z$. Put

$$P=Z\times Z,K=\beta Z\times \beta Z.$$

Then K is an infinite separable compactification of P, so $dens\left(K\right)=\omega$. Since both copies of Z are non-compact, Theorem 5.1 (iii) gives

$$d_K^{\partial }\left(P\right)=dens\left(P\right)\ge dens\left(Z\right)>\omega .$$

□

6. Compactification Spectra

For the boundary-approach spectrum ${\Sigma }_{\partial }\left(X\right)$, defined in Definition 2.2, order monotonicity identifies the upper endpoint. For non-compact locally compact Tychonoff spaces, the lower endpoint is attained by the one-point compactification.

Proposition 6.1. 

Let X be a non-compact locally compact Tychonoff space, and let $\alpha X=X\cup \{\infty \}$ be its one-point compactification. Then

$$min{\Sigma }_{\partial }\left(X\right)=d_{\alpha }^{\partial }\left(X\right),max{\Sigma }_{\partial }\left(X\right)=d_{\beta }^{\partial }\left(X\right),$$

and

$$d_{\alpha }^{\partial }\left(X\right)=min\left\{\right|D|:D\subset X\mathrm{and}{cl}_{X}D\mathrm{is}\mathrm{non}-\mathrm{compact}\}.$$

Proof. 

A locally compact dense subspace of a Hausdorff space is open. Thus X is open in every Hausdorff compactification $bX$, and collapsing the compact set $bX\setminus X$ to one point gives a continuous map $bX\to \alpha X$ fixing X. Hence

$$\alpha X\le bX\le \beta X,$$

and Proposition 4.1 gives the two endpoint identities.

A neighborhood of ∞ in $\alpha X$ is the complement of a compact subset of X. Therefore $\infty \in {cl}_{\alpha X}D$ if and only if D is not contained in any compact subset of X, which is equivalent to ${cl}_{X}D$ being non-compact. □

The lower endpoint is characterized by $\omega$-boundedness.

Corollary 6.2. 

Let X be a non-compact locally compact Tychonoff space. Then

$$d_{\alpha }^{\partial }\left(X\right)=\omega$$

if and only if X is not ω-bounded.

Proof. 

By the definition of $\omega$-boundedness recalled in Section 2 and Proposition 6.1, $d_{\alpha }^{\partial }\left(X\right)$ is the least cardinality of a subset of X with non-compact closure. Since this cardinal is infinite, it equals $\omega$ exactly when some countable subset of X has non-compact closure. □

Non-$\omega$-bounded clopen pieces give a sufficient condition for interval realization. No separability assumption is needed. It suffices that each piece contain a countable subset with non-compact closure, equivalently, that each piece fail to be $\omega$-bounded.

Theorem 6.3. 

Let X be a non-compact locally compact Tychonoff space and suppose that

$$X=\coprod _{\xi <\tau }{G}_{\xi },$$

where τ is an infinite cardinal and every $G_{\xi }$ is a non-empty clopen subspace that is not ω-bounded. Then there is a family of Hausdorff compactifications $b_{\lambda }X$, indexed by the infinite cardinals $\lambda \le \tau$, such that

$$d_{b_{\lambda }}^{\partial }\left(X\right)=\lambda$$

for every such λ, and

$$\omega \le \lambda \le \mu \le \tau ⟹b_{\lambda }X\le b_{\mu }X.$$

The first member may be chosen as $b_{\omega }X=\alpha X$.

If, in addition, $\tau =dens\left(X\right)$, then

$${\Sigma }_{\partial }\left(X\right)=\{\lambda :\omega \le \lambda \le dens\left(X\right)\}.$$

When $dens\left(X\right)>\omega$, the top member of the chain may be chosen as $b_{dens\left(X\right)}X=\beta X$.

Proof. 

Since the $G_{\xi }$ are non-empty open subspaces, every dense subset of X meets each $G_{\xi }$, and hence $\tau \le dens\left(X\right)$. For each $\xi <\tau$, choose a countably infinite set $A_{\xi }\subset G_{\xi }$ whose closure in $G_{\xi }$ is non-compact. This is possible because $G_{\xi }$ is not $\omega$-bounded. Since each $G_{\xi }$ is clopen in X, the closure of $A_{\xi }$ in X is the same non-compact set.

Identify cardinals with their initial ordinals. For the infinite cardinals $\lambda \le \tau$, set $I_{\omega }=⌀$ and, for $\lambda >\omega$, set $I_{\lambda }=\lambda$. Then $|I_{\lambda }|=\lambda$ for $\lambda >\omega$ and $I_{\lambda }\subseteq I_{\mu }$ whenever $\lambda \le \mu$. Each $G_{\xi }$ is non-compact, since every compact space is $\omega$-bounded. For $\xi \in I_{\lambda }$, let

$$S_{\xi }=\alpha G_{\xi }=G_{\xi }\cup \left\{p_{\xi }\right\}$$

be the one-point compactification of $G_{\xi }$. For $\xi \notin I_{\lambda }$, leave $G_{\xi }$ uncompactified. Put

$$Y_{\lambda }=\left(\coprod _{\xi \in I_{\lambda }}{S}_{\xi }\right)\bigsqcup \left(\coprod _{\xi \notin I_{\lambda }}{G}_{\xi }\right),b_{\lambda }X=\alpha Y_{\lambda }.$$

Write ${\infty }_{\lambda }$ for the point added in the one-point compactification $\alpha Y_{\lambda }$. The space $Y_{\lambda }$ is non-compact, locally compact and Hausdorff. Since the copy of X is dense in each compactified component $S_{\xi }$ and contains every uncompactified component, it is dense in $Y_{\lambda }$. It also meets infinitely many components of $Y_{\lambda }$ and therefore is not contained in any compact subset of $Y_{\lambda }$. Thus ${\infty }_{\lambda }$ lies in the closure of X, and $b_{\lambda }X$ is a compactification of X.

For $\lambda =\omega$, take $b_{\omega }X=\alpha X$. The set $A_0$ is countable and has non-compact closure in X, so Proposition 6.1 gives $d_{\alpha }^{\partial }\left(X\right)=\omega$.

Suppose $\lambda >\omega$. The remainder of X in $b_{\lambda }X$ is

$$\{p_{\xi }:\xi \in I_{\lambda }\}\cup \left\{{\infty }_{\lambda }\right\}.$$

If $D\subset X$ accumulates at every point of the remainder, then $p_{\xi }\in {cl}_{b_{\lambda }X}D$ forces $D\cap G_{\xi }\ne ⌀$ for every $\xi \in I_{\lambda }$. Since the $G_{\xi }$ are pairwise disjoint,

$$\left|D\right|\ge \lambda .$$

For the reverse inequality, set

$$D_{\lambda }=\bigcup _{\xi \in I_{\lambda }}{A}_{\xi }.$$

Then $|D_{\lambda }|=\lambda$. Because $A_{\xi }$ is not contained in any compact subset of $G_{\xi }$, the point $p_{\xi }$ lies in ${cl}_{S_{\xi }}{A}_{\xi }$ for every $\xi \in I_{\lambda }$. Moreover, $D_{\lambda }$ meets infinitely many components of the topological sum $Y_{\lambda }$, whereas every compact subset of a topological sum meets only finitely many components. Thus $D_{\lambda }$ is not contained in any compact subset of $Y_{\lambda }$, and

$${\infty }_{\lambda }\in {cl}_{b_{\lambda }X}{D}_{\lambda }.$$

Consequently $d_{b_{\lambda }}^{\partial }\left(X\right)=\lambda$.

It remains to verify coherence in the compactification order. If $\lambda \le \mu$, define

$$q_{\mu \lambda }:b_{\mu }X⟶b_{\lambda }X.$$

The map fixes X and each $p_{\xi }$ for $\xi \in I_{\lambda }$, sends $p_{\xi }$ to ${\infty }_{\lambda }$ for $\xi \in I_{\mu }\setminus I_{\lambda }$, and sends ${\infty }_{\mu }$ to ${\infty }_{\lambda }$. Continuity at points of X is immediate because X is open in both compactifications and $q_{\mu \lambda }{|}_X$ is the identity. At each retained point $p_{\xi }$ with $\xi \in I_{\lambda }$, the map is likewise the identity on the corresponding compactified component $S_{\xi }$. If $\xi \in I_{\mu }\setminus I_{\lambda }$ and U is a neighborhood of ${\infty }_{\lambda }$, then its complement C is compact in $Y_{\lambda }$ and $C\cap G_{\xi }$ is compact in $G_{\xi }$. A neighborhood of $p_{\xi }$ in $S_{\xi }$ avoiding $C\cap G_{\xi }$ maps into U. Continuity at ${\infty }_{\mu }$ follows from the same compact-complement argument. A compact $C\subset Y_{\lambda }$ meets only finitely many components, and its inverse image in each corresponding component of $Y_{\mu }$ is compact. Hence $q_{\mu \lambda }^{-1}\left(C\right)$ is compact in $Y_{\mu }$, so $q_{\mu \lambda }^{-1}\left(U\right)$ is a neighborhood of ${\infty }_{\mu }$. Thus $q_{\mu \lambda }$ is continuous and fixes X, proving $b_{\lambda }X\le b_{\mu }X$.

Finally, assume $\tau =dens\left(X\right)$. Every compactification has boundary-approach density between $\omega$ and $dens\left(X\right)$, while the preceding construction realizes every cardinal in this interval. Hence the displayed spectrum identity holds. If $dens\left(X\right)>\omega$, the constructed top member has value $dens\left(X\right)$ and lies below $\beta X$ in the compactification order. Order monotonicity and the general upper bound give

$$dens\left(X\right)\le d_{\beta }^{\partial }\left(X\right)\le dens\left(X\right),$$

so $d_{\beta }^{\partial }\left(X\right)=dens\left(X\right)$. We may therefore replace the constructed top member by $\beta X$ without disturbing the chain. □

Corollary 6.4. 

Let X be a non-compact locally compact Tychonoff space that is a topological sum of non-empty separable clopen subspaces. Then

$${\Sigma }_{\partial }\left(X\right)=\{\lambda :\omega \le \lambda \le dens\left(X\right)\},$$

and all values are realized along a chain in the compactification order. If $dens\left(X\right)>\omega$, the chain may be chosen with endpoints $\alpha X$ and $\beta X$.

Proof. 

Put $\kappa =dens\left(X\right)$. If $\kappa =\omega$, every boundary-approach density is infinite because X is non-compact, and (1) shows that each is at most $\omega$. Hence the spectrum is $\left\{\omega \right\}$. Assume $\kappa >\omega$ and write

$$X=\coprod _{j\in J}{X}_j$$

with every $X_j$ non-empty, clopen and separable. Every dense subset of X meets each $X_j$, while the union of countable dense subsets of the $X_j$ is dense in X. Hence $\left|J\right|=\kappa$. Partition J into $\kappa$ pairwise disjoint countably infinite sets $J_{\xi }$ and put

$$G_{\xi }=\coprod _{j\in J_{\xi }}{X}_j(\xi <\kappa ).$$

Each $G_{\xi }$ is clopen, locally compact, separable and non-compact. A countable dense subset of $G_{\xi }$ has non-compact closure, so $G_{\xi }$ is not $\omega$-bounded. Apply Theorem 6.3 with $\tau =\kappa$. □

Corollary 6.5. 

Every non-compact locally compact metrizable space X satisfies

$${\Sigma }_{\partial }\left(X\right)=\{\lambda :\omega \le \lambda \le dens\left(X\right)\},$$

and all these values can be realized along an order-coherent chain. If $dens\left(X\right)>\omega$, the chain may be chosen with endpoints $\alpha X$ and $\beta X$.

Proof. 

Every locally compact metrizable space is locally separable because each point has an open neighborhood with compact metrizable closure and hence a separable neighborhood. By the classical Alexandroff decomposition theorem for locally separable metrizable spaces ([23], 4.4.F(c)), X is a topological sum of non-empty separable clopen subspaces. Apply Corollary 6.4. □

Corollary 6.6. 

Let $X_1,\dots ,X_n$ be non-empty locally compact metrizable spaces, where $n\ge 2$ and at least two of the $X_i$ are non-compact. Put

$$P={\displaystyle \prod _{i=1}^n}{X}_i.$$

Then, for arbitrary Hausdorff compactifications $b_i{X}_i$,

$$d_{{\prod }_{i=1}^n{b}_i{X}_i}^{\partial }\left(P\right)=dens\left(P\right)=max{\Sigma }_{\partial }\left(P\right).$$

Moreover,

$${\Sigma }_{\partial }\left(P\right)=\{\lambda :\omega \le \lambda \le dens\left(P\right)\}.$$

Hence, if $dens\left(P\right)>\omega$, every infinite cardinal below $dens\left(P\right)$ is realized by some Hausdorff compactification of P, but by no factorwise product compactification ${\prod }_{i=1}^n{b}_i{X}_i$.

Proof. 

The finite product P is locally compact, metrizable and non-compact. Since at least two factors are non-compact, Theorem 5.1 gives the displayed product value. Corollary 6.5 gives the spectrum identity and identifies its maximum with $dens\left(P\right)$. □

Corollary 6.7. 

If X is a discrete space of infinite cardinality κ, then

$${\Sigma }_{\partial }\left(X\right)=\{\lambda :\omega \le \lambda \le \kappa \},$$

and all these values are realized along a chain in the compactification order. In particular,

$$d_{\alpha }^{\partial }\left(X\right)=\omega ,d_{\beta }^{\partial }\left(X\right)=\kappa .$$

7. Concluding Remarks and Further Questions

Boundary-approach density is an embedding-sensitive cardinal measuring the least size of a single subset of X whose closure reaches the entire remainder of the chosen compactification. The closed-core theorem converts this ambient condition into an internal compactness obstruction. In the Stone–Čech case, that obstruction is expressed entirely through complete separation in X. The order and mapping results give the corresponding monotonicity principles. For factorwise products, the dependence on the chosen factor compactifications disappears once at least two factors are non-compact. For finite products of locally compact metrizable spaces in this regime, the product theorem and the full-spectrum result together show that factorwise product compactifications attain only the maximum spectral value. When $dens\left(P\right)>\omega$, smaller infinite values are still realized, but only by non-factorwise compactifications. By contrast, the interval-realization theorem shows that if a non-compact locally compact Tychonoff space has an infinite clopen partition into non-$\omega$-bounded pieces, then an interval of boundary-approach values can be realized by compactifications chosen coherently in the compactification order. No metrizability or separability is needed for this construction. A separable clopen decomposition is a convenient sufficient hypothesis, and every locally compact metrizable space admits one ([23], 4.4.F(c)).

Ordinary compactification density behaves differently. For a fixed space, it is the same in every compactification [10], whereas the boundary-approach spectrum may be a full cardinal interval. Corollary 5.3 also shows that boundary-approach density can exceed the ordinary density of the ambient compactification itself.

These product, order, and spectrum results suggest the following three questions.

(Q1)

For which non-compact locally compact Tychonoff spaces X is

$${\Sigma }_{\partial }\left(X\right)=\{\lambda :d_{\alpha }^{\partial }\left(X\right)\le \lambda \le d_{\beta }^{\partial }\left(X\right)\}?$$

(Q2)

For which non-compact Tychonoff spaces X can all values in ${\Sigma }_{\partial }\left(X\right)$ be realized along a single chain in the compactification order?

(Q3)

Let $P={\prod }_{i\in I}{X}_i$ be a product of non-empty Tychonoff spaces with at least two non-compact factors and $dens\left(P\right)>\omega$. When does ${\Sigma }_{\partial }\left(P\right)$ contain a value strictly below $dens\left(P\right)$? By Theorem 5.1, every compactification realizing such a value must be non-factorwise.