| Title:
|
Properties of one-point completions of a noncompact metrizable space (English) |
| Author:
|
Henriksen, M. |
| Author:
|
Janos, L. |
| Author:
|
Woods, R. G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
1 |
| Year:
|
2005 |
| Pages:
|
105-123 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
metrizable |
| Keyword:
|
metric extensions and completions |
| Keyword:
|
completely metrizable |
| Keyword:
|
one-point metric extensions |
| Keyword:
|
extension traces |
| Keyword:
|
zerosets |
| Keyword:
|
clopen sets |
| Keyword:
|
Stone-Čech compactification |
| Keyword:
|
$\beta X\backslash X$ |
| Keyword:
|
hedgehog |
| MSC:
|
54D35 |
| MSC:
|
54E35 |
| MSC:
|
54E45 |
| MSC:
|
54E50 |
| idZBL:
|
Zbl 1121.54048 |
| idMR:
|
MR2175863 |
| . |
| Date available:
|
2009-05-05T16:49:57Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119512 |
| . |
| Reference:
|
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| Reference:
|
[B74] Bel'nov V.K.: Some theorems on metric extensions.Trans. Moscow Math. Soc. 30 (1974), 119-138. MR 0385812 |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
[W74] Woods R.G.: Zero-dimensional compactifications of locally compact spaces.Canadian J. Math. 26 (1974), 920-930. Zbl 0287.54023, MR 0350699 |
| . |
