Equivalence of Definitions of Closure of Topological Subspace

Theorem
Let $T$ be a topological space.

Let $H \subseteq T$.

The following definitions for the closure of $H$ in $T$ are equivalent:


 * $(1): \quad H^-$ is the union of $H$ and its limit points
 * $(2): \quad H^- := \displaystyle \bigcap_{H \subseteq K \subseteq T: K \text{ closed}} K$
 * $(3): \quad H^-$ is the smallest closed set that contains $H$
 * $(4): \quad H^-$ is the union of $H$ and its boundary
 * $(5): \quad H^-$ is the union of all isolated points of $H$ and all limit points of $H$

$(1) \iff (2)$
This is proved in Set Closure as Intersection of Closed Sets.

$(2) \iff (3)$
This is proved in Set Closure is Smallest Closed Set.