On (non-Menger) spaces whose closed nowhere dense subsets are Menger

TL;DR

This paper introduces od-Menger spaces, explores their properties, and constructs a specific example of a hereditarily Lindelöf, od-Menger, non-Menger space under CH, expanding understanding of topological space classifications.

Contribution

It defines od-Menger spaces, investigates their characteristics, and provides a construction of a non-Menger space with specific properties under the Continuum Hypothesis.

Findings

Existence of a hereditarily Lindelöf, od-Menger, non-Menger space under CH

Characterization of od-Menger spaces and their distinction from Menger spaces

Construction of a space with specific topological properties

Abstract

A space $X$ is od-Menger if it satisfies $\mathsf{U_{fin}}(\Delta_X, \mathcal{O}_X)$, where $\mathcal{O}_X,\Delta_X$ are the collection of covers of $X$ by respectively open subsets and open dense subsets. We show that under CH, there is a refinement of the usual topology on a subset of the reals which yields a hereditarily Lindel\"of, od-Menger, non-Menger, $0$-dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.