information:classification, information:extraction
A topology on a set

Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation X may be used to denote a set X endowed with the particular topology τ.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (i.e., its complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open.