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 (ε(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* = ßX\ X, and let Z(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 (ε(X),≤) and (ε(Y ),≤) are order-isomorphic iff Z(X*) and Z(Y*) are order-isomorphic, and iff X* and Y * are homeomorphic. We construct an order preserving bijection λ: ε(X) → Z(X*) such that a one-point completion in ε(X) is locally compact iff its image under λ 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.
© 2005 Charles University in Prague
Henriksen, Melvin, L. Janos, and R. G. Woods. "Properties of one-point completions of a noncompact metrizable space." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 105-123.