On $π$-compatible topologies and their special cases

Vitalij A. Chatyrko

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arXiv · 2308.12799

On $π$-compatible topologies and their special cases

Vitalij A. Chatyrko

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Abstract

Topologies $τ, σ$ on a set $X$ are called $π$-compatible if $τ$ is a $π$-network for $σ$, and vice versa. If topologies $τ, σ$ on a set $X$ are $π$-compatible then the families of nowhere dense sets (resp. meager sets or sets possessing the Baire property) of the spaces $(X, τ)$ and $(X, σ)$ coincide. A topology $σ$ on a set $X$ is called an admissible extension of a topology $τ$ on $X$ if $τ\subseteq σ$ and $τ$ is a $π$-network for $σ$. It turns out that examples of admissible extensions were occurred in literature several times. In the paper we provide some new facts about the $π$-compatibility and the admissible extension as well as about their particular cases.

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Vitalij A. Chatyrko. 2023-08-24. On $π$-compatible topologies and their special cases. https://arxiv.org/abs/2308.12799

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