Analysis of a topological basis problem
Recall that a subclass \(\mathcal{B}\) of a class \(\mathcal{K}\) of topological spaces is a basis provided each member of \(\mathcal{K}\) contains a member of \(\mathcal{B}\) as a subspace. This paper works toward a small basis for the class of uncountable, regular, first countable spaces, assuming PFA. Given a topological space \((X,\tau)\) and a collection \(\mathcal{C}\) of subsets of \(X\) two new topologies, called inner and outer, induced by \(\mathcal{C}\) are defined on \(X\): for example the inner, \(\tau^{I,\mathcal{C}}\) has base \(\{\{x\}\cup O^{I,\mathcal{C}}\ /\ x\in O\in\tau\}\), where \(O^{I,\mathcal{C}}=\cup\{C\in\mathcal{C}\ /\ C\subset O\}\). A couple of sample results assuming PFA: if \(X\) is regular and the inner topology is hereditarily Lindelöf then \(X\) either has a countable network or contains an uncountable Sorgenfrey subset; if \(X\) has cardinality \(\aleph_1\), is first countable and Hausdorff and the inner topology induced by a countable collection is hereditarily Lindelöf then \(X\) contains an uncountable subset that is metrisable or Sorgenfrey. There are also results involving perfect preimages and D-spaces.
scientific article; zbMATH DE number 7614055
scientific article