Topological Space

Topological spaces are

The Open and Closed Set Axioms

Let $X$ be a set, and $\tau$ a collection of subsets of $X$ (which will be called the open subsets of $X$ with respect to the topology $\tau$) verifying the following axioms:

1. $\varnothing \in \tau$
2. $X\in \tau$
3. Any finite intersection of sets in $\tau$ is again in $\tau$; i.e., if $U_{1},\ldots ,U_{n}\in \tau$, then $U_{1}\cap \ldots \cap U_{n}\in \tau$.
4. Any union of a family of sets $\{U_{\lambda }\}_{\lambda \in \Lambda }\subseteq \tau$ is in $\tau$; i.e., $\bigcup _{\lambda \in \Lambda }U_{\lambda }\in \tau$.

When these axioms are satisfied, we say that $(X,\tau )$ is a topological space of open sets $\tau$.

The Neighborhood Axioms

One can phrase a set of axioms for the definition of a topological space by defining the neighborhoods of points in that space. This is particularly useful when one considers topologies on topological abelian groups and topological rings by subgroups or ideals, respectively, because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of $0$.

Examples

1. Metric spaces