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
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