#### Continuous functions

A very simple definition of continuity exists for topological spaces. It nicely generalizes the definition from standard calculus. Let denote a function between topological spaces and . For any set , let the preimage of be denoted and defined by

 (4.4)

Note that this definition does not require to have an inverse.

The function is called continuous if is an open set for every open set . Analysis is greatly simplified by this definition of continuity. For example, to show that any composition of continuous functions is continuous requires only a one-line argument that the preimage of the preimage of any open set always yields an open set. Compare this to the cumbersome classical proof that requires a mess of 's and 's. The notion is also so general that continuous functions can even be defined on the absurd topological space from Example 4.4.

Steven M LaValle 2012-04-20