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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.
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