On Fodor's theorem: A topological version
We develop a generalization of Alexandroff and Fodor theorems in a topological way. For instance let x be a point of a space X, and \({\mathfrak V}(x)\) be the family of neighborhoods of x. We say that x is an Alexandroff point iff for every family \(x_ V\) for \(V\in {\mathfrak V}(x)\) satisfying \(x_ V\not\in V\) there are \(W\in {\mathfrak V}(x)\) and a basis \({\mathfrak B}\) of neighborhoods of x such that \(x_ V\not\in W\) for every V in \({\mathfrak B}\). By defining the notion of closed basis and stationary basis we have the notion of Fodor point. We prove that if \(\kappa_ 1,...,\kappa_ p\), \(p<\omega\) and \(\kappa_ k>\omega\) regular, setting \(X_ k=\kappa_ k+1=\kappa_ k\cup \{x_ k\}\) with the interval topology, then in \(\prod \{X_ k,\quad 1\leq k\leq p\}\) the point \(x=(x_ 1,...,x_ p)\) is a Fodor point and thus Alexandroff point. One of the questions is to know if in a \(T_ 2\) space Y the point \(y\in Y\) is in the closure of a countable set, then, can y be Alexandroff? (This cannot occur if y is a countable limit in \(T_ 2\)-space.) In particular: is it possible that \(p\in \beta {\mathbb{N}}\) can be Alexandroff? An ultrafilter \({\mathfrak U}\) on \(\omega ={\mathbb{N}}\) is Alexandroff iff for every family \((x_ A)_{A\in {\mathfrak U}}\) satisfying \(x_ A\not\in A\), there are \(W\in {\mathfrak U}\) and a basis \({\mathfrak B}\) of \({\mathfrak U}\) such that \(x_ A\not\in W\) for \(A\in {\mathfrak B}\). (This means \({\mathfrak U}\) is Alexandroff in \(\omega\) \(\cup \{{\mathfrak U}\})\). Is there an Alexandroff ultrafilter? Is there an ultrafilter which is not Alexandroff?
scientific article; zbMATH DE number 3897794
scientific article