Topological Space

Definition

A topological space is an ordered pair $(X, \tau)$ where $X$ is a set and $\tau \subseteq \mathcal{P}(X)$ is a collection of subsets of $X$ (called open sets) satisfying:

1. $\emptyset \in \tau$ and $X \in \tau$
2. $\tau$ is closed under arbitrary unions: if $\{U_\alpha\}_{\alpha \in A} \subseteq \tau$, then $\bigcup_{\alpha \in A} U_\alpha \in \tau$
3. $\tau$ is closed under finite intersections: if $U_1, \dots, U_n \in \tau$, then $\bigcap_{i=1}^n U_i \in \tau$

The collection $\tau$ is called a topology on $X$.

Examples

• Discrete topology: $\tau = \mathcal{P}(X)$ — every subset is open
• Indiscrete topology: $\tau = \{\emptyset, X\}$ — only the empty set and $X$ are open
• Metric topology: the topology induced by a metric, where open sets are unions of open balls

Related Concepts

• Metric Space — every metric space induces a topological space
• Metric — a distance function that generates a topology
• Continuity — defined via preimages of open sets