Question

Show that a subspace of regular space is regular.

Answer 1

Regular and Hausdorff spaces are very nicely behaved with respect to the product and subspace operations. ... Suppose that Y is a subspace of regular space X. Then if y ∈ Y , {y} = Y ∩ {y}, a closed set in the subspace topology as one-point subsets are closed in X. Thus one-point sets are closed in the subspace topology.