Equivalence of Definitions of Closure of Topological Subspace

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

Let $H^-$ denote the closure of $H$.

Then:


 * $(1): \quad H^- = \displaystyle \bigcap \left\{{K \supseteq H: K}\right.$ is closed in $\left.{T}\right\}$
 * $(2): \quad H^-$ is the smallest closed set that is a superset of $H$
 * $(3): \quad H^-$ is the union of $H$ and the boundary of $H$
 * $(4): \quad H^-$ is the union of all isolated points of $H$ and all limit points of $H$

$(1)$
This is proved in Set Closure as Intersection of Closed Sets.

$(2)$
This is proved in Set Closure is Smallest Closed Set.