Abstract
Recall that for a normal Hausdorff space X, we have \(\dim X = \dim \beta X\) and \( \mathop {\mathrm {Ind}} \nolimits X = \mathop {\mathrm {Ind}} \nolimits \beta X\). Moreover, a T 1 zero-dimensional space X can be embedded in a Cantor cube and so X has a zero-dimensional compactification. It is therefore natural to ask whether every normal Hausdorff space X has a compactification Y with \( \mathop {\mathrm {ind}} \nolimits Y = \mathop {\mathrm {ind}} \nolimits X\). In this chapter we construct a Hausdorff, perfectly normal space X with \( \mathop {\mathrm {ind}} \nolimits X= 1\) such that \(\dim Y = \mathop {\mathrm {ind}} \nolimits Y= \infty \) for every compactification Y of X. The first such example is due to van Mill and Przymusiński (Topol Appl 13:133–136, 1982). Bear in mind that by Proposition 5.3, \(\dim Y \leq \mathop {\mathrm {ind}} \nolimits Y\) for every Lindelöf space Y .
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References
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Charalambous, M.G. (2019). No Compactification Theorem for the Small Inductive Dimension of Perfectly Normal Spaces. In: Dimension Theory. Atlantis Studies in Mathematics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-030-22232-1_26
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DOI: https://doi.org/10.1007/978-3-030-22232-1_26
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