Subset of Metric Space is Closed iff contains all Zero Distance Points

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $H \subseteq A$.


Then $H$ is closed in $M$ if and only if:

$\forall x \in A: \map d {x, H} = 0 \implies x \in H$

where $\map d {x, H}$ denotes the distance between $x$ and $H$.


Proof

Necessary Condition

Let $H$ be closed in $M$.

Let $x \in A: \map d {x, H} = 0 \implies x \in H$.

By Existence of Sequence in Subset of Metric Space whose Limit is Infimum there exists a sequence $\sequence {a_n}$ of points of $H$ such that:

$\ds \lim_{n \mathop \to \infty} \map d {x, a_n} = 0$

So every neighborhood of $x$ contains points of $H$.

If some $a_n \in H$ then $x \in H$.

Otherwise each $a_n$ is different from $x$

Then $x$ is the limit of $\sequence {a_n}$ and hence of $H$.

Thus $x \in H$ by definition of closed set.

$\Box$


Sufficient Condition

Suppose $H$ is such that:

$\forall x \in A: \map d {x, H} = 0 \implies x \in H$

From Point at Zero Distance from Subset of Metric Space is Limit Point or Element:

$\map d {x, H} = 0$

if and only if either:

$x$ is a limit point of $H$
$x \in H$

Let $x$ be limit point of $H$.

Then by hypothesis $x \in H$.

That is:

$H$ contains all its limit points

So by definition $H$ is a closed set of $M$.

$\blacksquare$


Also see

If $H$ is closed in $M$, then $\map d {x, H} = 0$ if and only if $x \in H$.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Subset_of_Metric_Space_is_Closed_iff_contains_all_Zero_Distance_Points&oldid=628719"