Neighborhood in Topological Space has Subset Neighborhood

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Theorem

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.


That is:

$\forall x \in S: \forall N \in \mathcal N_x: \exists N' \in \mathcal N_x, N' \subseteq N: \forall y \in N': N' \in \mathcal N_y$

where $\mathcal N_x$ is the neighborhood filter of $x$.


Proof

By definition of neighborhood:

$\exists U \in \tau: x \in U \subseteq N \subseteq S$

where $U$ is an open set of $T$.

By Set is Open iff Neighborhood of all its Points, $N' = U$ fulfils the conditions of the statement.

$\blacksquare$


Sources