Continuity & Convergence

Definition (Continuity)

A mapping $f:X\to Y$ 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 $f:X\to Y$ is said to be a homeomorphism, if $f$ is bijective and both $f$ and the inverse mapping $f^{-1}:Y\to X$ are continuous.

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