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

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.