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, Tietzeâ€“Urysohn for normal spaces and equicontinuity

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