Equivalent Definitions for Topological Closure

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 $H$

Proof

$(1) \iff (2)$

This is proved in Set Closure as Intersection of Closed Sets.

$\blacksquare$

$(2) \iff (3)$

This is proved in Set Closure is Smallest Closed Set.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors