Closed Ball in Metric Space is Closed Neighborhood

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

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.

$\blacksquare$