Let \(X\) and \(Y\) be topological spaces and \(f\colon X\to Y\) a map. The map \(f\) is a closed map if \(f(A)\) is closed in \(Y\) for every closed \(A\subseteq X\), and an open map if \(f(U)\) is open in \(Y\) for every open \(U\subseteq X\) [@Mun00].
Neither condition implies or is implied by continuity, and neither implies the other.
The projection \(\pi_1\colon\RR\times\RR\to\RR\), \((x,y)\mapsto x\), is continuous and open but not closed: the hyperbola \(\ts{(x,y)\st xy=1}\) is closed in \(\RR\times\RR\), and its image \(\RR\setminus\ts{0}\) is not closed in \(\RR\) [@Mun00].
Let \(f\colon X\to Y\) be a continuous map of topological spaces.
If \(f\) is surjective and open or closed, then \(f\) is a quotient map; a quotient map need not be open or closed [@Mun00].
If \(X\) is compact and \(Y\) is Hausdorff, then \(f\) is closed [@Mun00].
If \(X\) is compact, \(Y\) is Hausdorff, and \(f\) is bijective, then \(f\) is a homeomorphism [@Mun00].