Definition:Topology

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

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


 * 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 $$\left({X, \vartheta}\right)$$ is called a topological space.

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

Alternative Definition
A more compact specification for the intersection of elements of $$\vartheta$$ can be given as:


 * 3. The intersection of any two elements of $$\vartheta$$ is an element of $$\vartheta$$.

It can be shown directly by induction that this specification is equivalent to no 3. above.

Sigma-Algebra
It is apparent from the definition of Sigma-Algebra that a topology and a sigma-algebra are tantamount to the same thing.