topology

Definition

Given set $X$, a topology on $X$ is a collection $\tau$ of subsets of $X$ called open subsets (open sets), where $\tau \subset P(X)$ (subset of power set) such that it is closed under forming

1. finite intersections (any intersection of finitely many elements of $\tau$ is an element of $\tau$)
2. arbitrary unions (any union of of elements of $\tau$ is a member of $\tau$)

where a topological space $(X,\tau)$ is a set $X$ equipped with such a topology.

It may be shown that the empty set $\emptyset$ and $X$ are elements of $\tau$, i.e. $\emptyset \in \tau$, $X \in \tau$.

References

1. https://ncatlab.org/nlab/show/topology
2. https://en.wikipedia.org/wiki/Topology
3. https://ncatlab.org/nlab/show/Introduction+to+Topology