Abstract
Every map T » UX from a set T to the underlying set UX of a compact Hausdorff space X admits a unique continuous extension βT » X from the Čech-Stone-compactification βT of T to X. Is it true for an arbitrary space X with this unique extension property to be already compact Hausdorff? No, there is a sophisticated counterexample [8]. Consequently, it makes sense to investigate the full subcategory of all such spaces in Top, say Comp β, which turns out to be reflective, containing compact Hausdorff spaces as reflective and bicoreflective subcategory. This paper deals with a new topological description of the spaces in Comp β, which yields more natural examples up to a finally dense class. Moreover, it turns out that there are very abstract categorical reasons for the concrete topological observations above.
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Richter, G. Reflective relatives of adjunctions. Appl Categor Struct 4, 31–41 (1996). https://doi.org/10.1007/BF00124112
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DOI: https://doi.org/10.1007/BF00124112