Definition:Topology

Definition
Let $X$ be any set and let $\vartheta$ be a collection of subsets of $X$.

Then $\vartheta$ is a topology on $X$ if:


 * $(1): \quad$ Any union of arbitrarily many elements of $\vartheta$ is an element of $\vartheta$;
 * $(2): \quad$ The intersection of any two elements of $\vartheta$ is an element of $\vartheta$;
 * $(3): \quad X$ is itself an element of $\vartheta$.

If $\vartheta$ is a topology on $X$, then $\left({X, \vartheta}\right)$ is called a topological space.

The elements of $\vartheta$ are called the open sets of $\left({X, \vartheta}\right)$.

Elementary Properties

 * In General Intersection Property of Topological Space, it is proved that a topology can equivalently be defined by the properties:


 * $(1): \quad$ Any union of arbitrarily many elements of $\vartheta$ is an element of $\vartheta$
 * $(2): \quad$ The intersection of any finite number of elements of $\vartheta$ is an element of $\vartheta$.


 * In Empty Set is Element of Topology it is shown that in any topological space $\left({X, \vartheta}\right)$ it is always the case that $\varnothing \in \vartheta$.

Also see

 * Sigma-Algebra, which looks similar on the surface to a topology, but closed under a countable number of unions. A topology has no such limitation on countability.