Topological Space

math topology

Definition

Topological Space

A topological space is a pair $(X,\tau )$, where $X$ is a set and $\tau$ is a collection of subsets of $X$ (called open sets) satisfying:

1. $\varnothing \in \tau$ and $X\in \tau$.
2. The union of any sub-collection of $\tau$ is in $\tau$.
3. The intersection of any finite sub-collection of $\tau$ is in $\tau$.

The collection $\tau$ is called a topology on $X$. Topological spaces provide the most general framework for defining concepts of continuity, convergence, and compactness.