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.