Properties of one-point completions of a noncompact metrizable space

Henriksen, Melvin, Janoš, Ludvík, and Woods, Grant R.. "Properties of one-point completions of a noncompact metrizable space." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 105-123. <http://eudml.org/doc/249571>.

@article{Henriksen2005,
abstract = {If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a 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 $T_\{1\}\le T_\{2\}$. If $X$ is noncompact and metrizable, then $(\mathcal \{M\} (X),\le )$ denotes the set of metric extensions of $X$, where $T_\{1\}$ and $T_\{2\}$ are identified if $T_\{1\}\le T_\{2\}$ and $T_\{2\}\le T_\{1\}$, i.e., if there is a homeomorphism of $T_\{1\}$ onto $T_\{2\}$ keeping $X$ pointwise fixed. $(\mathcal \{M\}(X),\le )$ is a large complicated poset studied extensively by V. Bel’nov [The structure of the set of metric extensions of a noncompact metrizable space, Trans. Moscow Math. Soc. 32 (1975), 1–30]. We study the poset $(\mathcal \{E\} (X),\le )$ 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 $\mathcal \{Z\}(X)$ be the poset of zerosets of $X$ partially ordered by set inclusion. TheoremIf $\,X$ and $Y$ are locally compact separable metrizable spaces, then $(\mathcal \{E\}(X),\le )$ and $(\mathcal \{E\} (Y),\le )$ are order-isomorphic iff $\,\mathcal \{Z\} (X^\{\ast \})$ and $\mathcal \{Z\}(Y^\{\ast \})$ are order-isomorphic, and iff $\,X^\{\ast \}$ and $Y^\{\ast \}$ are homeomorphic. We construct an order preserving bijection $\lambda : \mathcal \{E\} (X)\rightarrow \mathcal \{Z\} (X^\{\ast \})$ such that a one-point completion in $\mathcal \{E\} (X)$ 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.},
author = {Henriksen, Melvin, Janoš, Ludvík, Woods, Grant R.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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; Stone-Čech compactification; ; hedgehog},
language = {eng},
number = {1},
pages = {105-123},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Properties of one-point completions of a noncompact metrizable space},
url = {http://eudml.org/doc/249571},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Henriksen, Melvin
AU - Janoš, Ludvík
AU - Woods, Grant R.
TI - Properties of one-point completions of a noncompact metrizable space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 105
EP - 123
AB - If a metrizable space $X$ is dense in a metrizable space $Y$, then $Y$ is called a 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 $T_{1}\le T_{2}$. If $X$ is noncompact and metrizable, then $(\mathcal {M} (X),\le )$ denotes the set of metric extensions of $X$, where $T_{1}$ and $T_{2}$ are identified if $T_{1}\le T_{2}$ and $T_{2}\le T_{1}$, i.e., if there is a homeomorphism of $T_{1}$ onto $T_{2}$ keeping $X$ pointwise fixed. $(\mathcal {M}(X),\le )$ is a large complicated poset studied extensively by V. Bel’nov [The structure of the set of metric extensions of a noncompact metrizable space, Trans. Moscow Math. Soc. 32 (1975), 1–30]. We study the poset $(\mathcal {E} (X),\le )$ 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 $\mathcal {Z}(X)$ be the poset of zerosets of $X$ partially ordered by set inclusion. TheoremIf $\,X$ and $Y$ are locally compact separable metrizable spaces, then $(\mathcal {E}(X),\le )$ and $(\mathcal {E} (Y),\le )$ are order-isomorphic iff $\,\mathcal {Z} (X^{\ast })$ and $\mathcal {Z}(Y^{\ast })$ are order-isomorphic, and iff $\,X^{\ast }$ and $Y^{\ast }$ are homeomorphic. We construct an order preserving bijection $\lambda : \mathcal {E} (X)\rightarrow \mathcal {Z} (X^{\ast })$ such that a one-point completion in $\mathcal {E} (X)$ 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.
LA - eng
KW - metrizable; metric extensions and completions; completely metrizable; one-point metric extensions; extension traces; zerosets; clopen sets; Stone-Čech compactification; $\beta X\backslash X$; hedgehog; Stone-Čech compactification; ; hedgehog
UR - http://eudml.org/doc/249571
ER -