Definition:Topology/Definition 1

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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:

\((O1)\)   $:$   The union of an arbitrary subset of $\tau$ is an element of $\tau$.             
\((O2)\)   $:$   The intersection of any two elements of $\tau$ is an element of $\tau$.             
\((O3)\)   $:$   $S$ is an element of $\tau$.             


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 $\left({S, \tau}\right)$.


Also see

  • Results about topologies can be found here.


Sources