Definition:Topology

Definition
Let $S$ be a set.

A topology on $S$ is a subset $\tau \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ that satisfies the open set axioms:

If $\tau$ is a topology on $S$, then $\left({S, \tau}\right)$ is called a topological space.

The elements of $\tau$ are called the open sets of $\left({S, \tau}\right)$.

Elementary Properties

 * In General Intersection Property of Topological Space, it is proved that a topology can equivalently be defined by the properties:


 * $(1): \quad$ The union of an arbitrary subset of $\tau$ is an element of $\tau$.
 * $(2): \quad$ The intersection of any finite subset of $\tau$ is an element of $\tau$.


 * In Empty Set is Element of Topology it is shown that in any topological space $\left({S, \tau}\right)$ it is always the case that $\varnothing \in \tau$.

Also see

 * Basis
 * $\sigma$-algebra, which looks similar on the surface to a topology, but closed (in the algebraic sense) under countable unions. A topology has no such limitation on countability.