Continuous Maps on Topological Spaces

Continuous Function

Let and be topological spaces and a function. is called continuous if for every open set in , is open in .
is called continuous at if, for any open neighborhood of in , the set is a (not necessarily open) neighborhood of in .

e.g.

• The identity map is continuous;
• Any constant map is continuous;
• The composition of two continuous maps is always continuous.
  

Theorem

Let and be topological spaces. Consider a function . The following are equivalent:

1. is continuous on ;
2. is continuous at every point in ;
3. is closed in for every closed set in ;
4. whenever for a basis of ;
5. whenever for a subbasis of ;
6. for all ;
   

Proof 2 is obvious; 3 comes from definition of closed sets; 4 and 5 are because preimages preserve unions and intersections. We prove 6 here. Suppose is continuous. Since is closed, is closed in . Note that , so , hence , thus . For the other direction, suppose for all . Let be a closed subset of . Then , thus , so is closed in . Therefore, is continuous.

Theorem

Let be a function between two topological spaces and . Assume that ,where is open in for each index . Then is continuous if and only if is continuous for every .

Open and Closed Map

A map is open if for every open set , is open in . Similarly, it is closed if for every closed set , is closed in .