Neighbourhood (mathematics) - Topology From Neighbourhoods

Topology From Neighbourhoods

The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.

A neighbourhood system on is the assignment of a filter (on the set ) to each in, such that

1. the point is an element of each in
2. each in contains some in such that for each in, is in .

One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.