Ultrapowers of topological spaces (Q2295656)

To every topological space \(X\), every base \(\mathcal B\) of \(X\) and every free ultrafilter \(p\) over the set \( \omega\) of natural numbers the author associates a topological space over the ultrapower of the set \(X\) and then takes an appropriate quotient \(Ult_p(X, \mathcal B)\). Under a suitable condition on \(\mathcal B\) and assuming a separation property, a space \(X\) embeds in \(Ult_p(X, \mathcal B)\), hence, iterating the construction \( \omega_1\) times, the author obtains a \(p\)-compact space. The construction is motivated by recent results by \textit{M. Hrušák} et al. [Trans. Am. Math. Soc. 374, No. 2, 1277--1296 (2021; Zbl 1482.22002)] and generalizes [\textit{M. Di Nasso} and \textit{M. Forti}, Proc. Am. Math. Soc. 134, No. 6, 1809--1818 (2006; Zbl 1096.03056)]. The salient feature of the construction of \(Ult_p(X, \mathcal B)\) is the dependency upon the base \(\mathcal B\). Letting \(\mathcal B\) vary and iterating \( \omega_1\) times the author obtains many \(p\)-compactifications, for example, the ``universal'' \(p\)-compactification \(\beta_p(X)\) of a space \(X\), that is, the smallest \(p\)-compact extension of \(X\) in its Čech-Stone compactification. Other compactifications which can be obtained in this way are the one-point compactification of a locally compact space, as well as every \(p\)-compactification of a discrete space. It is left as an open problem whether every \(p\)-compactification of every space can be obtained in this way. In the final section the author studies those topological groups whose ultrapowers are topological groups. The author shows that ``there are severe restrictions to the use of ultrapowers to obtain \(p\)-compactifications of topological groups while preserving the property of being a topological group'', in particular, for abelian topological groups ``the condition of being precompact is necessary and that of being of bounded torsion is almost necessary.'' For a precompact abelian topological group of bounded torsion the universal \(p\)-compactification derived from the Weil completion can be obtained as the special case of the author's construction.