- What does it mean for \(E\subseteq X\) to be a dense subspace?
Solution
A subset \(E\subseteq X\) is dense in \(X\) if \[ \overline E=X. \] Equivalently, every nonempty open subset of \(X\) meets \(E\).
A point \(x\) lies in \(\overline E\) exactly when every open neighborhood of \(x\) intersects \(E\). Hence \(\overline E=X\) exactly when every nonempty open set intersects \(E\).