Let be a topological space. A subset N of X containing is said to be neighbourhood of if there exist an open set U containing such that N contains U, 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 X are neighbourhood of, because, we can find an open set such that
and
On the other hand, X is the only neighbourhood of, because, we can find the open set X such that
As another example, let with topology then are neighbourhood of. Similarly, are neighbourhoods of, and X 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 X 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 any two neighbourhoods of a point is also its neighbourhood in a topological space.

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

If A is a neighbourhood of point x, and, then show that B is also neighbourhood of point x.

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

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 M of a topological space X which contains a member of N(x) also belongs to N(x).

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