Properties of one-point completions of a noncompact metrizable space

Henriksen, M. ; Janos, L. ; Woods, R. G.

Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 105-123.

Résumé

If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a {\it metric extension\/} of $X$. If $T_{1}$ and $T_{2}$ are metric extensions of $X$ and there is a continuous map of $T_{2}$ into $T_{1}$ keeping $X$ pointwise fixed, we write $T_{1}\leq T_{2}$. If $X$ is noncompact and metrizable, then $(\Cal M (X),\leq)$ denotes the set of metric extensions of $X$, where $T_{1}$ and $T_{2}$ are identified if $T_{1}\leq T_{2}$ and $T_{2}\leq T_{1}$, i.e., if there is a homeomorphism of $T_{1}$ onto $T_{2}$ keeping $X$ pointwise fixed. $(\Cal M(X),\leq)$ is a large complicated poset studied extensively by V. Bel'nov [{\it The structure of the set of metric extensions of a noncompact metrizable space\/}, Trans. Moscow Math. Soc. {\bf 32} (1975), 1--30]. We study the poset $(\Cal E (X),\leq)$ of one-point metric extensions of a locally compact metrizable space $X$. Each such extension is a (Cauchy) completion of $X$ with respect to a compatible metric. This poset resembles the lattice of compactifications of a locally compact space if $X$ is also separable. For Tychonoff $X$, let $X^{\ast}=\beta X\backslash X$, and let $\Cal Z(X)$ be the poset of zerosets of $X$ partially ordered by set inclusion. \newline {\bf Theorem} {\sl If $\,X$ and $Y$ are locally compact separable metrizable spaces, then $(\Cal E(X),\leq)$ and $(\Cal E (Y),\leq)$ are order-isomorphic iff $\,\Cal Z (X^{\ast})$ and $\Cal Z(Y^{\ast})$ are order-isomorphic, and iff $\,X^{\ast}$ and $Y^{\ast}$ are homeomorphic\/}. We construct an order preserving bijection $\lambda : \Cal E (X)\rightarrow \Cal Z (X^{\ast})$ such that a one-point completion in $\Cal E (X)$ is locally compact iff its image under $\lambda$ is clopen. We extend some results to the nonseparable case, but leave problems open. In a concluding section, we show how to construct one-point completions geometrically in some explicit cases.

Détail

Classification : 54D35, 54E35, 54E45, 54E50
Keywords: metrizable; metric extensions and completions; completely metrizable; one-point metric extensions; extension traces; zerosets; clopen sets; Stone-Čech compactification; $\beta X\backslash X$; hedgehog