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 dispe...

Let Y be an absolute neighbourhood retract (ANR) for the class of metric spaces and let X be a Hausdorff space. Let Y X denote the space of continuous maps from X to Y equipped with the compact open topology. It is shown that if X is a CW complex then Y X is an ANR for the class of metric spaces if and only if Y X is metrizable and has the homotopy type of a CW complex. The same holds also when...

Let X be a Hausdorff topological space, Q(X,R) the space of all quasicontinuous functions on with values in R and ?UC topology uniform convergence compacta. If is hemicompact, then (Q(X,R), ?UC) metrizable thus many cardinal invariants, including weight, density cellularity coincide ?UC). We find further conditions under which these invariants as well characterizations some It known that weight...

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake systematic study those lcs such that (E, w) (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). following results obtained in our play key role: for every barrelled E, the (linearly Eberlein–Grothendieck) if only metrizable (E normable, respectively). main applications concern to continuo...