Question

Let U and V be open dense subsets of X. Prove that U intersection V is also dense in X.

Answer 1

Answer on Question #42314 – Math – Topology

Question. Let $U$ and $V$ be open dense subsets of $X$. Prove that $U$ intersection $V$ is also dense in $X$.

Proof. Recall that a subset $U\subset X$ is *dense* if for every non-empty open $W\subset X$ the intersection $U\cap W\neq\varnothing$.

Now let $W\subset X$ be any non-empty open subset. We should prove that $W\cap(U\cap V)\neq\varnothing$. Since $U$ is dense in $X$, we have that

$W\cap U\neq\varnothing.$

But $W\cap U$ is open as an intersection of two open subsets. Therefore

$(W\cap U)\cap V=W\cap(U\cap V)\neq\varnothing,$

since $V$ is dense as well.

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