Continuity

Entry conditions

Use topological continuity only when you have topological spaces on both domain and codomain.

Definitions

A function $f:X \to Y$ between topological spaces is continuous if for every open set $U$ in $Y$, the preimage $f^{-1}(U)$ is open in $X$.

Vocabulary (plain language)

• Preimage: the set of all points in $X$ that map into a set in $Y$.

Symbols used

• $f^{-1}(U)$: preimage of $U$ under $f$.

Intuition

Continuity is defined by the behavior of open sets, not by distances. If openness is preserved under pullback, the function is continuous.

Worked example

Every function from a discrete space is continuous because all subsets are open, so every preimage is open.

How to recognize the structure

• You can compute preimages.
• Preimages of opens are open.

Common mistakes

• Using distance-based continuity without specifying a metric.