Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 2 / 4E (Metric and Topological Spaces)

For topological spaces, continuity means that $f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$. Equivalently, inverse images of closed sets are closed. A homeomorphism is a bijection for which both the map and its inverse are continuous.

False. The constant continuous function $f:\mathbb{R}\to \mathbb{R}$, $f(x)=0$, sends the open set $\mathbb{R}$ to $\{0\}$, which is not open. Therefore continuity alone does not imply an open map.

False. Give $X=\mathbb{R}$ the discrete topology and $Y=\mathbb{R}$ its usual topology. The identity bijection $f:X\to Y$ is continuous because every inverse image is a subset of the discrete space and hence is open. But $f^{-1}:Y\to X$ is not continuous: the open set $\{0\}$ of $X$ has inverse image $\{0\}$, not open in $Y$. Thus a continuous bijection need not be a homeomorphism.

True. Let $g=f^{-1}$. For every open $U\subseteq X$,which is open in $Y$ because $f$ is an open map. Thus $g$ is continuous. Together with the given continuity and bijectivity of $f$, this establishes a homeomorphism. This is inverse continuity from an open bijection.