General topology grew out of a number of areas, most importantly the following:
General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.
A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space.
Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations.
Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.
Every sequence of points in a compact metric space has a convergent subsequence.
Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.
The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.