Continuous images preserve compactness, connectedness, and separability.Suggest a slogan
Let \(f\colon X\to Y\) be continuous.
If \(X\) is compact, then \(f(X)\) is compact [@Mun00].
If \(X\) is connected, then \(f(X)\) is connected [@Mun00].
If \(D\subseteq X\) is dense, then \(f(D)\) is dense in \(f(X)\) with the subspace topology; in particular, if \(X\) is separable, then so is \(f(X)\), and if \(f\) is surjective, then \(f(D)\) is dense in \(Y\).
A continuous map need not be open or closed: the constant map \(\RR\to\RR\), \(x\mapsto 0\), sends the open set \(\RR\) to \(\ts{0}\), which is not open, and \(x\mapsto e^x\) sends the closed set \(\RR\) to \((0,\infty)\), which is not closed in \(\RR\).