13.1 Topological spaces
The results of the previous section show that many important results concerning metric spaces depend only on the topology. We now generalize this, by introducing the notion of a topological space. This is traditionally defined in terms of open sets. A topological space is a set X, together with a collection τ of subsets of X which satisfy:
•the empty set and X are in τ ;
•if then ;
•if O1 and O2 are in τ then O1 ∩ O2 τ.
Then τ is the topology on X, and the sets in τ are called open sets. The conditions ...