Continuity in Topological Spaces

Continuous means preimages of open sets are open.

The idea

Continuity in a topological space restates the epsilon-delta definition from calculus using open sets alone, so that it makes sense in a space where nothing is measured.

The epsilon-delta condition says that for each tolerance demanded around $f(x)$, some tolerance around $x$ maps inside it. A tolerance around a point is an open set containing it, so the condition asks, at each point $x$, for an open set around $x$ that $f$ carries into a given open set around $f(x)$. Demanding this at every point at once compresses the whole condition into one statement about preimages.

A map $f \colon X \to Y$ between topological spaces is continuous when, for every open set $V \subseteq Y$, the preimage $f^{-1}(V) = \{\, x \in X : f(x) \in V \,\}$ is open in $X$. On metric spaces this selects exactly the maps that epsilon-delta continuity selects; on all other spaces it is the definition of continuity.

The definition runs backwards, from open sets in the target to open sets in the source. The forward condition — that images of open sets are open — is a different property, called being an open map, and a continuous map need not have it.

Ways to work on it

• Walkthrough. The preimage-of-open-is-open definition and the closed-set form.
• Practice. Decide whether a stated condition certifies continuity.
• Hardest. Test a concrete map against the definition with an open set.

Not sure where to start? Take the ten-question placement test.