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- W1566830734 abstract "If a metrizable space X is dense in a metrizable space Y , then Y is called a metric extension of X. If T1 and T2 are metric extensions of X and there is a continuous map of T2 into T1 keeping X pointwise fixed, we write T1 � T2. If X is noncompact and metrizable, then (M(X),�) denotes the set of metric extensions of X, where T1 and T2 are identified if T1 � T2 and T2 � T1, i.e., if there is a homeomorphism of T1 onto T2 keeping X pointwise fixed. (M(X),�) 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 (E(X),�) 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� = �XX, and letZ(X) be the poset of zerosets of X partially ordered by set inclusion. Theorem If X and Y are locally compact separable metrizable spaces, then (E(X),�) and (E(Y ),�) are order-isomorphic iff Z(X � ) andZ(Y � ) are order-isomorphic, and iff Xand Yare homeomorphic. We construct an order preserving bijection � :E(X)! Z(X � ) such that a one-point completion inE(X) is locally compact iff its image under � 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." @default.
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- W1566830734 date "2005-01-01" @default.
- W1566830734 modified "2023-09-28" @default.
- W1566830734 title "Properties of One-Point Completions of a Noncompact Metrizable Space" @default.
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