Preservation and Destruction in Simple Refinements

If σ is a topology on X and A ⊆ X, we let 〈σ,A〉 denote the topology generated by σ and A, i.e., the topology with σ ∪ {A} as a subbasis. Any refinement of a topology obtained like this – by declaring just one new set to be open – we call simple. The present paper investigates the preservation of various properties in simple refinements. The locally closed sets (sets open in their closure) play a crucial role here: it turns out that many properties are preserved in a simple refinement by A if and only if A is locally closed. We prove this for the properties of regularity, completely regularity, (complete) metrizability, and (complete) ultrametrizability. We also show that local compactness is preserved in a simple refinement by A if and only if both A and its complement are locally closed.