General Topology

• Definition, characterisations
• Constructions
• Bases
• Continuity continuity and constructions
• Separation separation and constructions
• Countability, density
• Compact spaces proper and continuous maps are closed, alexandroff and stone-cech compactifications, tychonoff, compactness and constructions, paracompactness and partition of unity
• Filters
• Connected spaces connectedness and constructions
• Miscellaneous spaces
• Nets
• Order topology and semicontinuity
• 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
• The compact-open topology
• Homotopy things which are invariant under homotopy
• Covering spaces proper local homeomorphisms are precisely finite covering maps
• Fiber bundles and fibrations
• CW complexes
• Simplicial complexes
• Pointed spaces and support

Sources

• Bourbaki, General Topology
• Grothendieck TVS
• Hatcher Algebraic Topology
• Lamotke Semisimpliziale algebraische Topologie