Category of Topological Spaces - Relationships To Other Categories

Relationships To Other Categories

• The category of pointed topological spaces Top_• is a coslice category over Top.
• The homotopy category hTop has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of Top. One can likewise form the pointed homotopy category hTop_•.
• Top contains the important category Haus of topological spaces with the Hausdorff property as a full subcategory. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with dense images in their codomains, so that epimorphisms need not be surjective.