| Neighbourhood (topology) |
Article Index for Neighbourhood |
Information AboutNeighbourhood (topology) |
|
In Topology and related areas of Mathematics , a neighbourhood (or '''neighborhood''') is one of the basic concepts in a Topological Space . Intuitively speaking, a neighbourhood of a point is a Set containing the point where you can "wiggle" or "move" the point a bit without leaving the set. This concept is closely related to the concepts of Open Set and Interior . DEFINITION If ''X'' is a Topological Space and ''p'' is a point in ''X'', a neighbourhood of ''p'' is a set ''V'', which contains an Open Set ''U'' containing ''p''. : Note that the neighbourhood ''V'' need not be an open set itself. If ''V'' is open it is called an open neighbourhood. Some authors require that neighbourhoods be open; be careful to note conventions. If ''S'' is a Subset of ''X'', a neighbourhood of ''S'' is a set ''V'', which contains an open set ''U'' containing ''S''. It follows that a set ''V'' is a neighbourhood of ''S'', if and only if, it is a neighbourhood of all the points in ''S''. The collection of all neighbourhoods of a point is called the Neighbourhood System at the point. IN A METRIC SPACE In a Metric Space ''M'' = (''X'',''d''), a set ''V'' is a neighbourhood of a point ''p'' if there exists an Open Ball with center ''p'' and radius ''r'', : which is contained in ''V''. ''V'' is called uniform neighbourhood of a set ''S'' if there exists a positive number ''r'' such that for all elements ''p'' of ''S'', : is contained in ''V''. EXAMPLES Given the set of Real Number s R with the usual Euclidean Metric and a subset ''V'' defined as : then ''V'' is a neighbourhood for the set N of Natural Number s, but is ''not'' a uniform neighbourhood of this set. 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 neighborhood system on ''X'' is the assignment of a Filter ''N(x)'' (on the set ''X'') to each ''x'' in ''X'', such that # the point ''x'' is an element of each ''U'' in ''N(x)'' # each ''U'' in ''N(x)'' contains some ''V'' in ''N(x)'' such that for each ''y'' in ''V'', ''U'' is in ''N(y)''. 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. UNIFORM NEIGHBOURHOODS In a to ''X'' \ ''V'', that is there exists no Entourage containing ''P'' and ''X'' \ ''V''. SIGNIFICANCE OF NEIGHBOURHOODS IN ANALYSIS OF REAL FUNCTIONS Neighbourhoods in more than one dimension are generally chosen equivalent to Euclidean metrics for their symmetry and readability. But for the analysis of functions, this choice is wholly arbitrary. Other notions of distance will (as they ought to) lead to the same results in analysis, if they are properly formulated. Consider, for instance, the following notion of distance in two dimensions: let be in the -neighbourhood of iff |
|
|