Basic Properties of Neighborhood in Topological Space

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Theorem

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


$N1$: Point in Topological Space has Neighborhood

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in S$.


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


$N2$: Point in Topological Space is Element of its Neighborhood

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in S$.

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


Then $a \in N$.


$N3$: Superset of Neighborhood in Topological Space is Neighborhood

Let $T = \left({S, \tau}\right)$ 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 $T$.


$N4$: Intersection of Neighborhoods in Topological Space is Neighborhood

Let $T = \left({S, \tau}\right)$ 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 $T$.


$N5$: Neighborhood in Topological Space has Subset Neighborhood

Let $T = \left({S, \tau}\right)$ 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