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 $T_{1}\leq T_{2}$. 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 [{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 $(\Cal E (X),\leq )$ 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 {Theorem} {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.
Keywords: metrizable, metric extensions and completions, completely metrizable, one-point metric extensions, extension traces, zerosets, clopen sets, Stone-\v {C}ech compactification, $\beta X\backslash X$, hedgehog
AMS Subject Classification: Primary 54E45, 54E50; Secondary 54E35, 54D35