Closed Ball in Metric Space is Closed Neighborhood
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon^-} x$ is a closed neighborhood of $x$ in $M$.
Proof
From Closed Ball contains Smaller Open Ball:
- $\map {B_\epsilon} x \subseteq \map {B_\epsilon^-} x$
where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.
Hence $\map {B_\epsilon^-} x$ is a neighborhood of $x$ by definition.
From Closed Ball is Closed in Metric Space:
- $\map {B_\epsilon^-} x$ is closed in $M$
It follows that $\map {B_\epsilon^-} x$ is a closed neighborhood of $x$ in $M$ by definition.
$\blacksquare$