Basic Properties of Neighborhood in Metric Space

Theorem

This page gathers together the basic properties of a neighborhood in a metric space.

$\text N 1$: Point in Metric Space has Neighborhood

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

Let $a \in A$ be a point in $M$.

Then there exists some neighborhood of $a$ in $M$.

$\text N 2$: Point in Metric Space is Element of its Neighborhood

Let $N$ be a neighborhood of $a$ in $M$.

Then $a \in N$.

$\text N 3$: Superset of Neighborhood in Metric Space is Neighborhood

Let $N$ be a neighborhood of $a$ in $M$.

Let $N \subseteq N' \subseteq A$.

Then $N'$ is a neighborhood of $a$ in $M$.

$\text N 4$: Intersection of Neighborhoods in Metric Space is Neighborhood

Let $N, N'$ be neighborhoods of $a$ in $M$.

Then $N \cap N'$ is a neighborhood of $a$ in $M$.

$\text N 5$: Neighborhood in Metric Space has Subset Neighborhood

Let $N$ be a neighborhood of $a$ in $M$.

Then there exists a neighborhood $N'$ of $a$ such that:

$(1): \quad N' \subseteq N$

$(2): \quad N'$ is a neighborhood of each of its points.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Theorem $4.8$