Jacobi integral: Difference between revisions

Latest revision as of 19:40, 5 January 2015

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-In [[topology]] and related areas of [[mathematics]], the '''neighbourhood system''' or '''neighbourhood filter''' <math>\mathcal{V}(x)</math> for a point ''x'' is the collection of all [[neighbourhood (mathematics)|neighbourhood]]s for the point ''x''.
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-A '''neighbourhood basis''' or '''local basis''' for a point ''x'' is a [[filter base]] of the neighbourhood filter, i.e. a subset
-:<math>\mathcal{B}(x) \subset \mathcal{V}(x)</math>
-such that
-:<math>\forall V \in \mathcal{V}(x) \quad \exists B \in \mathcal{B}(x) \mbox{ with } B \subset V</math>.
-That is, for any neighbourhood <math>V</math> we can find a neighbourhood <math>B</math> in the neighbourhood basis which is contained in <math>V</math>.
-Conversely, as with any filter base, the local basis allows to get back the corresponding neighbourhood filter as <math>\mathcal{V}(x) =\left\{ V \supset B~:~ B \in \mathcal{B}(x)\right\}</math>.<ref>Stephen Willard, ''General Topology'' (1970) Addison-Wesley Publishing (''See Chapter 2, Section 4'')</ref>
-==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, <math>\mathcal{V}(x) = \{ X \}</math>
-*In a [[metric space]], for any point ''x'', the sequence of [[open ball]]s around ''x'' with radius 1/''n'' form a [[countable]] neighbourhood basis <math>\mathcal{B}(x) = \{ B_{1/n}(x) ; n \in \mathbb N^* \}</math>. 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 <math>\nu</math> is given by
-: <math> \left \{ \mu \in \mathcal{M}(E) : | \mu f_i - \nu f_i | < \varepsilon_i , i=1,\ldots, n\right \} </math>
-where <math>f_i</math> are continuous bounded functions from ''E'' to the real numbers.
-==Properties==
-In a [[semi normed space]], that is a [[vector space]] with the [[topology]] induced by a [[semi norm]], all neighbourhood systems can be constructed by [[Translation (geometry)|translation]] of the neighbourhood system for the point 0,
-:<math>\mathcal{V}(x) = \mathcal{V}(0) + x .</math>
-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]].
-Every neighbourhood system for a non empty set ''A'' is a [[filter (mathematics)|filter]] called the [[neighbourhood filter]] for ''A''.
-==See also==
-*[[Neighbourhood (mathematics)|neighbourhood]]
-*[[Base (topology)]]
-*[[Locally convex topological vector space]]
-*[[Filter (mathematics)]]
-==References==
-<references/>
-{{DEFAULTSORT:Neighbourhood System}}
-[[Category:General topology]]

Jacobi integral: Difference between revisions

Latest revision as of 19:40, 5 January 2015

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