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

Mathieu Baillif, Santi Spadaro

TL;DR

This paper investigates od-Menger spaces, defined by $\mathsf{U_{fin}}(\Delta_X, \mathcal{O}_X)$, and examines when such spaces fail to be Menger. It establishes structural constraints for od-Menger spaces, showing, for example, that an od-Menger Lindelöf space with dense $M(X)$ is necessarily Menger, and it analyzes how subspaces and closures behave under od-Menger assumptions. The authors construct CH-based examples yielding Lindelöf, $0$-dimensional, first-countable spaces that are od-Menger but not Menger, via Luzin subspaces inside Pixley–Roy-type spaces like $\mathcal{K}(\mathbb{P})$ and related spaces, and discuss related results including Sakai’s $K_B$ construction. The results highlight sharp distinctions between od-Menger and Menger properties, including product behavior and the limits of using unions of small subfamilies to preserve Menger-type properties. The work also places these phenomena in the broader context of Luzin spaces, $G_\delta$ diagonals, and set-theoretic assumptions such as CH and MA.

Abstract

A space $X$ is od-Menger if it satisfies $\mathsf{U_{fin}}(Δ_X, \mathcal{O}_X)$, where $\mathcal{O}_X,Δ_X$ 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 Lindelöf, od-Menger, non-Menger, $0$-dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.

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

TL;DR

This paper investigates od-Menger spaces, defined by , and examines when such spaces fail to be Menger. It establishes structural constraints for od-Menger spaces, showing, for example, that an od-Menger Lindelöf space with dense is necessarily Menger, and it analyzes how subspaces and closures behave under od-Menger assumptions. The authors construct CH-based examples yielding Lindelöf, -dimensional, first-countable spaces that are od-Menger but not Menger, via Luzin subspaces inside Pixley–Roy-type spaces like and related spaces, and discuss related results including Sakai’s construction. The results highlight sharp distinctions between od-Menger and Menger properties, including product behavior and the limits of using unions of small subfamilies to preserve Menger-type properties. The work also places these phenomena in the broader context of Luzin spaces, diagonals, and set-theoretic assumptions such as CH and MA.

Abstract

A space is od-Menger if it satisfies , where are the collection of covers of 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 Lindelöf, od-Menger, non-Menger, -dimensional, first countable space. We also investigate the properties of spaces which are od-Menger but not Menger.
Paper Structure (3 sections, 26 theorems, 7 equations)

This paper contains 3 sections, 26 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Properties of spaces that are od-Menger but non-Menger
  3. Examples of $\mathsf{U_{fin}}\bigl({\Delta},{\mathcal{O}}\bigr) \not\Longrightarrow \mathsf{U_{fin}}\bigl({\mathcal{O}},{\mathcal{O}}\bigr)$

Key Result

Lemma 1.1

Let "has dM" and "has dR" stand for "has a dense Menger subspace" and "has a dense Rothberger subspace", respectively. The implications below hold for any space $X$.

Theorems & Definitions (33)

  • Lemma 1.1
  • Theorem 1.2: Mills, Wattel and Blair
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 1.5: Folklore, probably
  • Lemma 1.6
  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 23 more