# Dense and open sets in R^n.

by MathematicalPhysicist
Tags: dense, sets
 P: 230 you can proove pretty simple like this: A set A is dense if and only if $B \cap A$ is nonempty for all non-empty open sets B, (at least that is the definition we used for dense). Then let U and V be open dense sets, you need to prove that $U\cap V$ is dense, so let let W be any open non-empty set. You need to show that $W \cap (U \cap V)$ is non-empty, but $W \cap (U \cap V) = (W \cap U) \cap V$ and because W and U are open $W \cap U$ is open, and because U is dense it is non-empty, then $(W \cap U) \cap V$ is a intersection of a non-empty open set, with V, and again this is dense, so that intersection must be non-empty. QED. edit: didn't see you needed countable, this only works for finite, sorry about that.