On the convergence and character spectra of compact spaces. (English) Zbl 1198.54003

The character spectrum \(\chi S(p,X)\) of a non-isolated point \(p\) in a space \(X\) is \(\{\chi(p,Y)\mid p\) is non-isolated in \(Y\subset X\}\), while \(\chi S(X)=\bigcup\{\chi S(x,X)\mid x\in X\) is non-isolated\(\}\). A number of properties of \(\chi S(X)\), and the related convergence spectrum \(cS(X)\), are discussed. In particular it is shown that if \(X\) is compact, if cardinal \(\lambda=\lambda^{<\widehat{t}(X)}\) and if \(p\in X\) has character \(>\lambda\) then \(\lambda\in\chi S(p,X)\). Also that adding \(\lambda\) Cohen reals to a model of GCH gives an extension such that for each \(\kappa\leq\lambda\) there is compact \(X\) with \(\chi S(X)=\{\omega,\kappa\}\).


54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54A35 Consistency and independence results in general topology
54D30 Compactness
Full Text: DOI