Topological properties preserved by weakly discontinuous maps and weak homeomorphisms

TL;DR

This paper investigates how weakly discontinuous maps and weak homeomorphisms preserve key topological properties and classifies certain zero-dimensional spaces up to weak homeomorphism.

Contribution

It introduces the concept of weakly discontinuous maps and weak homeomorphisms, demonstrating their property-preserving capabilities and providing a classification of infinite zero-dimensional $\sigma$-Polish spaces.

Findings

Weak homeomorphisms preserve network weight, hereditary Lindelöf number, and dimension.

Classification of infinite zero-dimensional $\sigma$-Polish metrizable spaces into 9 types.

Any such space is weakly homeomorphic to one of the listed nine spaces.

Abstract

A map $f:X\to Y$ between topological spaces is called weakly discontinuous if each subspace $A\subset X$ contains an open dense subspace $U\subset A$ such that the restriction $f|U$ is continuous. A bijective map $f:X\to Y$ between topological spaces is called a weak homeomorphism if $f$ and $f^{-1}$ are weakly discontinuous. We study properties of topological spaces preserved by weakly discontinuous maps and weak homeomorphisms. In particular, we show that weak homeomorphisms preserve network weight, hereditary Lindel\"of number, dimension. Also we classify infinite zero-dimensional $\sigma$-Polish metrizable spaces up to a weak homeomorphism and prove that any such space $X$ is weakly homeomorphic to one of 9 spaces: $\omega$, $2^\omega$, $\mathbb N^\omega$, $\mathbb Q$, $\mathbb Q\oplus 2^\omega$, $\mathbb Q\times 2^\omega$, $\mathbb Q\oplus\mathbb N^\omega$, $(\mathbb Q\times 2^\omega)\oplus\mathbb N^\omega$, $\mathbb Q\times\mathbb N^\omega$.