Topological spaces with an $\omega^\omega$-base

TL;DR

This paper investigates topological spaces with an $oldsymbol{ extomega^oldsymbol{ extomega}}$-base, revealing their similarities to first-countable and generalized metric spaces, and exploring their properties and cardinality bounds.

Contribution

It introduces the concept of $oldsymbol{ extomega^oldsymbol{ extomega}}$-bases in topological spaces and analyzes their properties, extending known results from first-countable and metric spaces.

Findings

Spaces with an $ extomega^ extomega$-base share properties with first-countable spaces.

Many bounds on the cardinality of first-countable spaces hold for $ extomega^ extomega$-based spaces.

Spaces with a locally uniform $ extomega^ extomega$-base exhibit properties typical of generalized metric spaces.

Abstract

Given a partially ordered set $P$ we study properties of topological spaces $X$ admitting a $P$-base, i.e., an indexed family $(U_\alpha)_{\alpha\in P}$ of subsets of $X\times X$ such that $U_\beta\subset U_\alpha$ for all $\alpha\le\beta$ in $P$ and for every $x\in X$ the family $(U_\alpha[x])_{\alpha\in P}$ of balls $U_\alpha[x]=\{y\in X:(x,y)\in U_\alpha\}$ is a neighborhood base at $x$. A $P$-base $(U_\alpha)_{\alpha\in P}$ for $X$ is called locally uniform if the family of entourages $(U_\alpha U_\alpha^{-1}U_\alpha)_{\alpha\in P}$ remains a $P$-base for $X$. A topological space is first-countable if and only if it has an $\omega$-base. By Moore's Metrization Theorem, a topological space is metrizable if and only if it is a $T_0$-space with a locally uniform $\omega$-base. In the paper we shall study topological spaces possessing a (locally uniform) $\omega^\omega$-base. Our results show that spaces with an $\omega^\omega$-base share some common properties with first countable spaces, in particular, many known upper bounds on the cardinality of first-countable spaces remain true for countably tight $\omega^\omega$-based topological spaces. On the other hand, topological spaces with a locally uniform $\omega^\omega$-base have many properties, typical for generalized metric spaces. Also we study Tychonoff spaces whose universal (pre- or quasi-) uniformity has an $\omega^\omega$-base and show that such spaces are close to being $\sigma$-compact.