In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x.
A neighbourhood basis or local basis for a point x is a filter base of the neighbourhood filter, i.e. a subset
such that
- .
That is, for any neighbourhood we can find a neighbourhood in the neighbourhood basis which is contained in .
Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as .
Read more about Neighbourhood System: Examples, Properties
Famous quotes containing the word system:
“We recognize caste in dogs because we rank ourselves by the familiar dog system, a ladderlike social arrangement wherein one individual outranks all others, the next outranks all but the first, and so on down the hierarchy. But the cat system is more like a wheel, with a high-ranking cat at the hub and the others arranged around the rim, all reluctantly acknowledging the superiority of the despot but not necessarily measuring themselves against one another.”
—Elizabeth Marshall Thomas. “Strong and Sensitive Cats,” Atlantic Monthly (July 1994)