# Basic Properties of Neighborhood in Topological Space

## Theorem

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

### $\text N 1$: Point in Topological Space has Neighborhood

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Then there exists in $T$ at least one neighborhood of $x$.

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

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Let $N$ be a neighborhood of $x$ in $T$.

Then $a \in N$.

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

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Let $N$ be a neighborhood of $x$ in $T$.

Let $N \subseteq N' \subseteq S$.

Then $N'$ is a neighborhood of $x$ in $T$.

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

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Let $M, N$ be a neighborhoods of $x$ in $T$.

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

### $\text N 5$: Neighborhood in Topological Space has Subset Neighborhood

Let $T = \struct {S, \tau}$ be a topological space.

Let $x \in S$.

Let $N$ be a neighborhood of $x$ in $T$.

Then there exists a neighborhood $N'$ of $x$ 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 $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Theorem $3.1$