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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
##### Keywords:
linearly ordered spaces; neighbourhood retract