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