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

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

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

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

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

**Neighbourhood System:**

Let be a topological space. The set of all neighbourhoods of a point is said to be a neighbourhood system of . It is denoted by . The above example shows that neighbourhood system.

**Theorems:**

• The topological space itself is a neighbourhood of each of its points.

• A subset of a topological space is open if and only if it is the neighbourhood of each of its own points.

• The intersection of two neighbourhoods of a point is also its neighbourhood in a topological space.

• The union of two neighbourhoods of a point is also its neighbourhood in a topological space.

• If is a neighbourhood of , and , then show that is also neighbourhood of .

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

• The neighbourhood system of a point is a non empty set.

• The intersection of a finite number of neighbourhoods of a point is also its neighbourhood.

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

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