# Neighborhood of a Point

Let be a topological space. A subset of containing is said to be the neighborhood of if there exists an open set containing such that contains , i.e.

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 with topology (known as a Sierpinski space), then and are neighborhoods of because we can find an open set such that

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

As another example, let with topology then are neighborhoods of . Similarly, are neighborhoods of , and is the only neighborhood of and . It is clear from this illustration that a point may have more than one neighborhood.

**Neighborhood System**

Let be a topological space. The set of all neighborhoods of a point is said to be a neighborhood system of . It is denoted by . The above example shows this neighborhood system.

**Theorems**

• The topological space 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 is a neighborhood of and , then show that is also a neighborhood of .

• If is a neighborhood of , then show that there exists an open set such that is also a neighborhood of and is a neighborhood of each point of .

• 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 of a topological space which contains a member of also belongs to .

• Each neighborhood of a point of a cofinite topological space is open.