Proper subspace inherited properties, new characterizations of classical topological properties, and related new properties (Q2928819)

This article is dedicated to the discussion of proper subspace inherited properties. It is shown that for a \(i=0, 1, 2, 3\) the separation axioms \(T_{i}\) are proper subspace inherited properties and for \(i=0, 1\) the separation axioms \(R_{i}\) are proper subspace inherited properties. It is also shown that weakly Urysohn, regular, normal, Lindelöf, compact, separable, first countable, and second countable are proper subspace inherited properties.