On one-point metrizable extensions of locally compact metrizable spaces. (English) Zbl 1117.54037
This paper can be considered as a continuation of the study of one-point metrizable extensions by M. Henriksen, L. Janos and R. G. Woods [Commentat. Math. Univ. Carol. 46, 105–123 (2005; Zbl 1121.54048)]. For a non-compact, metrizable space \(X\), \({\mathcal E}(X)\) denotes the set of all one-point metrizable extensions of \(X\) and \({\mathcal E}_K(X)\) denotes the set of all locally compact elements of \({\mathcal E}(X)\). The set \({\mathcal E}(X)\) can be partially ordered by the partial order \(\leq\) defined by, for \(Y_1, Y_2\in{\mathcal E}(X)\), \(Y_1\leq Y_2\) if there exists a continuous map from \(Y_2\) to \(Y_1\) which keeps \(X\) pointwise fixed. M. Henriksen, L. Janos and R. G. Woods proved in [op. cit.] that, for all locally compact non-compact separable metrizable spaces \(X\) and \(Y\), \({\mathcal E}(X)\) and \({\mathcal E}(Y)\) are order-isomorphic if and only if \(X^\ast=\beta X\setminus X\) and \(Y^\ast=\beta Y\setminus Y\) are homeomorphic. They also defined a map \(\lambda\) from \({\mathcal E}(X)\) to the set \({\mathcal Z}(X^\ast)\) of all zero-sets in \(X^\ast\) by \(\lambda(Y)=\bigcap_{n<\omega}\text{cl}_{\beta X}(U_n\cap X)\setminus X\), where \(Y=X\cup\{p\}\in{\mathcal E}(X)\) and \(\{U_n:n<\omega\}\) is an open base at \(p\) in \(Y\).
In this paper, answering their question, the author proves that for every locally compact non-compact metrizable space \(X\), \(\lambda({\mathcal E}(X))\) consists of exactly those non-empty zero-sets of \(\beta X\) which miss \(X\), and \(\lambda({\mathcal E}_K(X))\) consists of exactly those elements of \(\lambda({\mathcal E}(X))\) which are open and closed in \(X^\ast\). The remainder of this paper is devoted to the study of the relation between \({\mathcal E}(X)\) and \({\mathcal E}_K(X)\) and their order structures. The author defines \({\mathcal E}_S(X)=\{X\cup\{p\}\in{\mathcal E}(X):p\) has a separable neighborhood in \(X\cup\{p\}\}\), and proves many results including the following ones: (i) For every locally compact non-compact metrizable space \(X\), \({\mathcal E}_K(X)\) and \({\mathcal E}(X)\) are never order-isomorphic; (ii) under the assumption of the continuum hypothesis, for every locally compact non-compact metrizable spaces \(X\) and \(Y\), if \(X^\ast\) and \(Y^\ast\) are homeomorphic, then \({\mathcal E}_S(X)\) and \({\mathcal E}_S(Y)\) are order-isomorphic.
In this paper, answering their question, the author proves that for every locally compact non-compact metrizable space \(X\), \(\lambda({\mathcal E}(X))\) consists of exactly those non-empty zero-sets of \(\beta X\) which miss \(X\), and \(\lambda({\mathcal E}_K(X))\) consists of exactly those elements of \(\lambda({\mathcal E}(X))\) which are open and closed in \(X^\ast\). The remainder of this paper is devoted to the study of the relation between \({\mathcal E}(X)\) and \({\mathcal E}_K(X)\) and their order structures. The author defines \({\mathcal E}_S(X)=\{X\cup\{p\}\in{\mathcal E}(X):p\) has a separable neighborhood in \(X\cup\{p\}\}\), and proves many results including the following ones: (i) For every locally compact non-compact metrizable space \(X\), \({\mathcal E}_K(X)\) and \({\mathcal E}(X)\) are never order-isomorphic; (ii) under the assumption of the continuum hypothesis, for every locally compact non-compact metrizable spaces \(X\) and \(Y\), if \(X^\ast\) and \(Y^\ast\) are homeomorphic, then \({\mathcal E}_S(X)\) and \({\mathcal E}_S(Y)\) are order-isomorphic.
Reviewer: Haruto Ohta (Shizuoka)
MSC:
| 54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |
| 54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |
| 54E35 | Metric spaces, metrizability |
| 54E45 | Compact (locally compact) metric spaces |
Keywords:
one-point extension; metrizable extension; Stone-Čech compactification; remainder; order-isomorphic; Parovičenko spaceCitations:
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