Definition:Topology

Definition
Let $S$ be a set such that $S \ne \varnothing$.

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:


 * 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 $\varnothing \in \tau$.

Note
Notwithstanding the result Empty Set Satisfies Topology Axioms, the stipulation that $S \ne \varnothing$ is standard in the literature.

However, this condition is often omitted in published texts, and taken for granted. When it is mentioned, it is usually given as an afterthought.

This website will follow tradition and always assume that the condition $S \ne \varnothing$ holds.

Also see

 * $\sigma$-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.