On meager function spaces, network character and meager convergence in topological spaces

TL;DR

This paper explores the relationship between network character, meager convergence, and function spaces in topological spaces, revealing new properties of points, sequences, and filters in these contexts.

Contribution

It establishes new connections between network character and meager convergence, and characterizes when function spaces are meager based on these properties.

Findings

Existence of non-isolated points with countable network character in compact Hausdorff spaces

Injective sequences with meager filter convergence at points of countable character

Meagerness of function spaces under certain convergence conditions

Abstract

For a non-isolated point $x$ of a topological space $X$ the network character $nw_\chi(x)$ is the smallest cardinality of a family of infinite subsets of $X$ such that each neighborhood $O(x)$ of $x$ contains a set from the family. We prove that (1) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with $nw_\chi(x)=\aleph_0$; (2) for each point $x\in X$ with countable character there is an injective sequence in $X$ that $\F$-converges to $x$ for some meager filter $\F$ on $\omega$; (3) if a functionally Hausdorff space $X$ contains an $\F$-convergent injective sequence for some meager filter $\F$, then for every $T_1$-space $Y$ that contains two non-empty open sets with disjoint closures, the function space $C_p(X,Y)$ is meager. Also we investigate properties of filters $\F$ admitting an injective $\F$-convergent sequence in $\beta\omega$.