Examples
- Trivially the neighbourhood system for a point is also a neighbourhood basis for the point.
- Given a space X with the indiscrete topology the neighbourhood system for any point x only contains the whole space,
- In a metric space, for any point x, the sequence of open balls around x with radius 1/n form a countable neighbourhood basis . This means every metric space is first-countable.
- In the weak topology on the space of measures on a space E, a neighbourhood base about is given by
where are continuous bounded functions from E to the real numbers.
Read more about this topic: Neighbourhood System
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