# Show that the preimage of dense set is dense?

by mathgirl313
Tags: dense, preimage
 P: 1 It is definitely false. Let ##X = \mathbb{R}## be the reals with the discrete topology, let ##Y = \mathbb{R}## be the reals with the Euclidean topology, and let $$f : X\to Y\\ x\mapsto x.$$ Then ##\mathbb{Q}## is dense in ##Y##, but ##\overline{f^{-1}(\mathbb{Q})} = \overline{\mathbb{Q}} = \mathbb{Q}## in X, because every set is closed in the discrete topology, and hence ##f^{-1}(\mathbb{Q})## is not dense in ##(\mathbb{R},\tau_{\textrm{disc}})##.