- What is the closure of a subspace \(E\subseteq X\)?
Solution
1. The closure \(\overline E\) is the intersection of all closed subsets of \(X\) containing \(E\).
Arbitrary intersections of closed sets are closed, so this is the unique smallest closed subset containing \(E\).
2. Equivalently,\[\boxed{\overline E=\{x\in X:U\cap E\ne\varnothing\text{ for every open neighborhood }U\ni x\}.}\]
A point fails to lie in the intersection from 1 exactly when it has an open neighborhood contained in the complement of some closed set containing \(E\), hence disjoint from \(E\).