Compact spaces proper and continuous maps are closed, alexandroff and stone-cech compactifications, tychonoff, compactness and constructions, paracompactness and partition of unity

Uniform spaces uniform (equi-)continuity, uniform completion, image of complete spaces in complete spaces, closed subspace of complete space is complete, Tietze–Urysohn for normal spaces and equicontinuity

Metric spaces pseudometrics (alexandroff in metric setting? or uniform?), metrizability theorem