# Continuity & Convergence

Definition (Continuity)

A mapping $f:X→Y$ between topological spaces $X$ and $Y$ is called

continuous, if for each open subset $V$ of $Y$ the inverse image $f_{−1}(V)$ is an open subset of $X$.

Slogan: continuous $=$ The inverse image of every open subset is open.

Definition (Homeomorphism)

Suppose $X$ and $Y$ are topological spaces. A mapping $f:X→Y$ is said to be a

homeomorphism, if $f$ is bijective and both $f$ and the inverse mapping $f_{−1}:Y→X$ are continuous.