Exercise 1, § 1

Let X be a topological space and Y a subspace of X. We say that Y is very dense in X if for every closed subset F of X, F is the closure of $F \cap Y$. a) For Y to be very dense in X, it is necessary and sufficient that for every nonempty locally closed subset F of X, $F \cap Y$ is nonempty. b) If Y is very dense in X and Z a subspace of Y very dense in X, Z is very dense in X. If Y is very dense in X and F a locally closed subset of X, $Y \cap F$ is very dense in F. c) If Y is very dense in X, one has $\dim(Y) = \dim(X)$.