A neighbourhood basis or local basis for a point x is a filter base of the neighbourhood filter, i.e. a subset
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 .
where are continuous bounded functions from E to the real numbers.
This is because, by assumption, vector addition is separate continuous in the induced topology. Therefore the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the topology is defined by a translation invariant metric or pseudometric.
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