Definition:Topological Space

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

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


 * 1) $$\varnothing, X \in \vartheta$$.
 * 2) Any union of arbitrarily many elements of $$\vartheta$$ is an element of $$\vartheta$$.
 * 3) Any intersection of finitely many elements of $$\vartheta$$ is an element of $$\vartheta$$.

If $$\vartheta$$ is a topology on $$X$$, then $$X$$ together with $$\vartheta$$ is called a topological space.