Topological Space - Specializations and Generalizations

Specializations and Generalizations

The following spaces and algebras are either more specialized or more general than the topological spaces discussed above.

• Proximity spaces provide a notion of closeness of two sets.
• Metric spaces embody a metric, a precise notion of distance between points.
• Uniform spaces axiomatize ordering the distance between distinct points.
• Cauchy spaces axiomatize the ability to test whether a net is Cauchy. Cauchy spaces provide a general setting for studying completions.
• Convergence spaces capture some of the features of convergence of filters.
• Grothendieck sites are categories with additional data axiomatizing whether a family of arrows covers an object. Sites are a general setting for defining sheaves.