# Neighborhood of a Point

Let $\left( {X,\tau } \right)$ be a topological space. A subset $N$ of $X$ containing $x \in X$ is said to be the neighborhood of $x$ if there exists an open set $U$ containing $x$ such that $N$ contains $U$, i.e.

$x \in U \subseteq X$

A neigborhood of a point is not necessarily an open set. However, if a neighborhood of a point is an open set, we call it an open neighborhood of that point.

If $X = \left\{ {a,b} \right\}$ with topology $\tau = \left\{ {\phi ,\left\{ a \right\},X} \right\}$ (known as a Sierpinski space), then $\left\{ a \right\}$ and $X$ are neighborhoods of $a$ because we can find an open set $\left\{ a \right\}$ such that

$a \in \left\{ a \right\} \subseteq \left\{ a \right\}$ and $a \in \left\{ a \right\} \subset X$

On the other hand, $X$ is the only neighborhood of $b$ because we can find the open set $X$ such that

$b \in X \subseteq X$

As another example, let $X = \left\{ {a,b,c,d} \right\}$ with topology $\tau = \left\{ {\phi ,\left\{ a \right\},\left\{ b \right\},\left\{ {a,b} \right\},X} \right\}$ then $\left\{ a \right\},\left\{ {a,b} \right\},\left\{ {a,c} \right\},\left\{ {a,d} \right\},\left\{ {a,b,c} \right\},\left\{ {a,c,d} \right\},X$ are neighborhoods of $a$. Similarly, $\left\{ b \right\},\left\{ {a,b} \right\},\left\{ {a,c} \right\},\left\{ {b,d} \right\},\left\{ {a,b,c} \right\},\left\{ {a,b,d} \right\},\left\{ {a,c,d} \right\},X$ are neighborhoods of $b$, and $X$ is the only neighborhood of $c$ and $d$. It is clear from this illustration that a point $x$ may have more than one neighborhood.

Neighborhood System

Let $\left( {X,\tau } \right)$ be a topological space. The set of all neighborhoods of a point $x \in X$ is said to be a neighborhood system of $x$. It is denoted by $N\left( x \right)$. The above example shows this neighborhood system.

Theorems
• The topological space $X$ itself is a neighborhood of each of its points.
• A subset of a topological space is open if and only if it is the neighborhood of each of its own points.
• The intersection of two neighborhoods of a point is also its neighborhood in a topological space.
• The union of two neighborhoods of a point is also its neighborhood in a topological space.
• If $A$ is a neighborhood of $x$ and $A \subset B$, then show that $B$ is also a neighborhood of $x$.
• If $A$ is a neighborhood of $x$, then show that there exists an open set $B$ such that $B$ is also a neighborhood of $x$ and $A$ is a neighborhood of each point of $B$.
• The neighborhood system of a point is a non empty set.
• The intersection of a finite number of the neighborhoods of a point is also its neighborhood.
• Any subset $M$ of a topological space $X$ which contains a member of $N(x)$ also belongs to $N(x)$.
• Each neighborhood of a point of a cofinite topological space is open.