Continuous functions

A very simple definition of continuity exists for topological spaces. It nicely generalizes the definition from standard calculus. Let $ f:
X \rightarrow Y$ denote a function between topological spaces $ X$ and $ Y$. For any set $ B \subseteq Y$, let the preimage of $ B$ be denoted and defined by

$\displaystyle f^{-1}(B) = \{ x \in X \;\vert\; f(x) \in B \} .$ (4.4)

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

The function $ f$ is called continuous if $ f^{-1}(O)$ is an open set for every open set $ O \subseteq Y$. 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 $ \delta$'s and $ \epsilon$'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