Baire spaces, k‐spaces, and some properly hereditary properties

Adnan Al-Bsoul

doi:10.1155/ijmms.2005.3697

Abstract

A topological property is properly hereditary property if whenever every proper subspace has the property, the whole space has the property. In this note, we will study some topological properties that are preserved by proper subspaces; in fact, we will study the following topological properties: Baire spaces, second category, sequentially compact, hemicompact, δ‐normal, and spaces having dispersion points. Also, we will solve some open problems raised by Al‐Bsoul (2003) and Arenas (1996) and conclude this note by some open problems.

Highlights

Puertas suggested the following question which was studied by Arenas in his paper [2]: (∗) if every proper subspace has the property, the whole space has the property
Arenas [2] proved that the topological properties Ti, i = 0,1,2,3, are properly hereditary properties
For Arenas, the property (∗) makes sense only for the topological properties that are hereditary to all subsets

Summary

Introduction

This is not the case; in this note we are going to give two topological properties which are not properly (resp., closed, open) hereditary properties. Let X be a space such that every proper subspace of X is a second category, and let {Gn : n ∈ N} be a countable family of open dense subsets of X. Is being a k-space a properly (closed) hereditary property?

Results

Conclusion