Definition:Topology

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

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


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

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