On order structure of the set of one-point Tychonoff extensions of a locally compact space

If a Tychonoff space $X$ is dense in a Tychonoff space $Y$, then $Y$ is called a Tychonoff extension of $X$. Two Tychonoff extensions $Y_1$ and $Y_2$ of $X$ are said to be equivalent, if there exists a homeomorphism $f:Y_1\rightarrow Y_2$ which keeps $X$ pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of $X$. We identify those extensions of $X$ which belong to the same equivalence classes. For two Tychonoff extensions $Y_1$ and $Y_2$ of $X$, we write $Y_2\leq Y_1$, if there exists a continuous function $f:Y_1\rightarrow Y_2$ which keeps $X$ pointwise fixed. This is a partial order on the set of all Tychonoff extensions of $X$. If a Tychonoff extension $Y$ of $X$ is such that $Y\backslash X$ is a singleton, then $Y$ is called a one-point extension of $X$. Let $T(X)$ denote the set of all one-point extensions of $X$. We study the order structure of the partially ordered set $(T(X),\leq)$. For a locally compact space $X$, we define an order-anti-isomorphism from $T(X)$ onto the set of all non-empty closed subsets of $\beta X\backslash X$. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of $X$, the set of all one-point Lindelof extensions of $X$, the set of all one-point pseudocompact extensions of $X$, and the set of all one-point Cech-complete extensions of $X$, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of $\beta X\backslash X$. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of $C^*(X)$ and the topology of subspaces of $\beta X\backslash X$.

💡 Research Summary

The paper investigates the partially ordered set of one‑point Tychonoff extensions of a locally compact space (X). An extension (Y) of (X) is a Tychonoff space containing (X) densely; two extensions are considered equivalent if there is a homeomorphism fixing every point of (X). The authors introduce a natural preorder: for extensions (Y_{1},Y_{2}) we write (Y_{2}\le Y_{1}) when there exists a continuous map (f:Y_{1}\to Y_{2}) that is the identity on (X). This relation is reflexive, antisymmetric (modulo the equivalence) and transitive, thus defining a partial order on the equivalence classes of extensions.

The focus is on one‑point extensions, i.e. extensions for which the remainder (Y\setminus X) consists of a single point. The collection of all such extensions is denoted by (T(X)). The central result is an order‑anti‑isomorphism between ((T(X),\le)) and the lattice of non‑empty closed subsets of the remainder (\beta X\setminus X) of the Stone–Čech compactification of (X). For a one‑point extension (Y) with added point (p), the map sends (Y) to the closure of ({p}) in (\beta X) intersected with (\beta X\setminus X). This correspondence is bijective and reverses the order: if (Y_{2}\le Y_{1}) then the associated closed set for (Y_{1}) is contained in that for (Y_{2}). Consequently, the order structure of one‑point extensions is completely captured by the topology of (\beta X\setminus X).

The authors then refine the picture by considering several natural subclasses of one‑point extensions:

Locally compact one‑point extensions (\mathcal{L}(X));
Lindelöf one‑point extensions (\mathcal{L!i}(X));
Pseudocompact one‑point extensions (\mathcal{P}(X));
Čech‑complete one‑point extensions (\mathcal{C}(X)).

Each subclass corresponds, under the anti‑isomorphism, to a distinguished family of closed subsets of (\beta X\setminus X) (e.g., complements of open sets, (G_{\delta})‑sets, zero‑sets, (F_{\sigma})‑sets). This yields precise inclusion relations among the subclasses: for instance (\mathcal{L}(X)\subseteq\mathcal{L!i}(X)\subseteq\mathcal{P}(X)\subseteq\mathcal{C}(X)), mirroring the corresponding inclusions of the associated families of closed sets.

Cardinality estimates are obtained by exploiting the size of (\beta X\setminus X). Whenever (\beta X\setminus X) is infinite, the lattice of its non‑empty closed subsets has cardinality at least (2^{\aleph_{0}}); consequently each of the families (\mathcal{L}(X),\mathcal{L!i}(X),\mathcal{P}(X),\mathcal{C}(X)) has cardinality at least continuum. In many cases (e.g., when (X) is non‑compact and not (\sigma)-compact) the lower bound can be sharpened to (|\beta X\setminus X|).

In the final section the authors connect these topological results with the algebraic structure of the Banach algebra (C^{}(X)) of bounded continuous real‑valued functions on (X). They consider the lattice (\mathcal{I}) of proper ideals of (C^{}(X)) and construct a map (\Psi:\mathcal{I}\to T(X)) that assigns to each ideal a corresponding one‑point extension. This map preserves the order in the opposite direction: (I_{1}\subseteq I_{2}) implies (\Psi(I_{2})\le\Psi(I_{1})). Composing (\Psi) with the anti‑isomorphism to closed subsets of (\beta X\setminus X) yields an order‑preserving bijection between the ideal lattice of (C^{*}(X)) and a sublattice of closed subsets of the Stone–Čech remainder. Thus the paper provides a unified framework that translates between three realms: extension theory, the topology of (\beta X\setminus X), and the ideal structure of function algebras.

Overall, the work demonstrates that for locally compact spaces the seemingly modest class of one‑point extensions carries a rich order structure that is completely reflected in the topology of the Stone–Čech remainder. By identifying precise anti‑isomorphisms and applying them to various subclasses and to (C^{*}(X)), the authors not only answer several classical questions about cardinalities and inclusion relations but also open a pathway for future investigations into multi‑point extensions, non‑locally‑compact bases, and deeper interactions between topology and functional analysis.