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Products of zero-dimensional compact lots as remainders. (English) Zbl 0822.54022
Many results have been obtained characterizing those spaces \(X\) which have a (Hausdorff) compactification \(\alpha X\) so that \(\alpha X-X\) is homeomorphic to a particular space \(Y\). Here the author is concerned with spaces \(Y\) which are products of compact zero-dimensional linearly ordered spaces and methods to construct compactifications of spaces \(X\) which have \(Y\) as a remainder. Necessary and sufficient conditions for when these constructions are possible are provided. One construction is done by directly topologizing the set \(X\cup Y\), while another is done by embedding \(X\) in a parallelotope. Using the latter technique, a characterization of those spaces \(X\) which have a compactification \(\alpha X\) such that \(\alpha X-X\) is a retract of \(X\) and \(\alpha X-X\) is homeomorphic to a product of compact zero-dimensional linearly ordered spaces is obtained. This result depends on a theorem by G. D. Faulkner [Proc. Am. Math. Soc. 103, No. 3, 984-988 (1988; Zbl 0649.54013)] which relates when a compact \(\alpha X-X\) is a retract of \(\alpha X\) to singular maps and compactifications. An analogous result for \(\alpha X-X\) being a neighbourhood retract is also presented.
This work uses the fact that compact zero-dimensional linearly ordered spaces are generalized double arrow spaces over some set. The notion of generalized double arrow spaces was developed in one of the author’s previous papers [Topology Proc. 17, 317-324 (1992; Zbl 0794.54030)].
MSC:
54D40 Remainders in general topology
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D30 Compactness
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