# Category:Definitions/Examples of Topologies

This category contains definitions of examples of topologies.

Let $S$ be a set.

A topology on $S$ is a subset $\tau \subseteq \powerset S$ of the power set of $S$ that satisfies the open set axioms:

 $(\text O 1)$ $:$ The union of an arbitrary subset of $\tau$ is an element of $\tau$. $(\text O 2)$ $:$ The intersection of any two elements of $\tau$ is an element of $\tau$. $(\text O 3)$ $:$ $S$ is an element of $\tau$.

If $\tau$ is a topology on $S$, then $\struct {S, \tau}$ is called a topological space.

The elements of $\tau$ are called the open sets of $\struct {S, \tau}$.

## Subcategories

This category has the following 35 subcategories, out of 35 total.