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In Topology and related areas of Mathematics a net or '''Moore-Smith sequence''' is a generalization of a Sequence , intended to unify the various notions of Limit and generalize them to arbitrary Topological Space s. Limits of nets accomplish for all topological spaces what limits of sequences accomplish for First-countable Space s such as Metric Space s. A sequence is usually indexed by the Natural Numbers which are a Totally Ordered Set . Nets generalize this concept by weakening the Order Relation on the index set to that of a Directed Set . Nets were first introduced by E. H. Moore and H. L. Smith in 1922 . An equivalent notion, called Filter , was developed in 1937 by Henri Cartan . DEFINITION If ''X'' is a topological space, a ''net'' in ''X'' is a Function from some Directed Set ''A'' to ''X''. If ''A'' is a directed set, we often write a net from ''A'' to ''X'' in the form (''x''α), which expresses the fact that the element α in ''A'' is mapped to the element ''x''α in ''X''. We usually use ≥ to denote the binary relation given on ''A''. EXAMPLES Since the Natural Number s with the usual order form a directed set and a sequence is a function on the natural numbers, every sequence is a net. Another important example is as follows. Given a point ''x'' in a topological space, let N''x'' denote the set of all Neighbourhood s containing ''x''. Then ''N''''x'' is a directed set, where the direction is given by reverse inclusion, so that ''S'' ≥ ''T'' if and only if ''S'' is contained in ''T''. For ''S'' in ''N''''x'', let ''x''''S'' be a point in ''S''. Then ''x''''S'' is a net. As ''S'' increases with respect to ≥, the points ''x''''S'' in the net are constrained to lie in decreasing neighbourhoods of ''x'', so intuitively speaking, we are led to the idea that ''x''''S'' must tend towards ''x'' in some sense. We can make this limiting concept precise. LIMITS OF NETS If (''x''α) is a net from a directed set ''A'' into ''X'', and if ''Y'' is a subset of ''X'', then we say that (''x''α) is ''eventually in'' ''Y'' if there exists an α in ''A'' so that for every β in ''A'' with β ≥ α, the point ''x''β lies in ''Y''. If (''x''α) is a net in the topological space ''X'', and ''x'' is an element of ''X'', we say that the net ''converges towards x'' or ''has limit x'' and write :lim ''x''α = ''x'' if and only if :for every Neighborhood ''U'' of ''x'', (''x''α) is eventually in ''U''. Intuitively, this means that the values ''x''α come and stay as close as we want to ''x'' for large enough α. Note that the example net given above on the Neighbourhood System of a point ''x'' does indeed converge to ''x'' according to this definition. EXAMPLES OF LIMITS OF NETS
SUPPLEMENTARY DEFINITIONS If ''D'' and ''E'' are directed sets, and ''h'' is a function from ''D'' to ''E'', then ''h'' is called cofinal if for every ''e'' in ''E'' there is a ''d'' in ''D'' so that if ''q'' is in ''D'' and ''q'' ≥ ''d'' then ''h''(''q'') ≥ ''e''. In other words, the Image ''h''(''D'') is Cofinal in ''E''. If ''D'' and ''E'' are directed sets, ''h'' is a cofinal function from ''D'' to ''E'', and φ is a net on set ''X'' based on ''E'', then φo''h'' is called a subnet of φ. All subnets are of this form, by definition. If φ is a net on ''X'' based on directed set ''D'' and ''A'' is a subset of ''X'', then φ is frequently in ''A'' if for every α in ''D'' there is a β in ''D'', β ≥ α so that φ(β) is in ''A''. A net φ on set ''X'' is called universal if for every subset ''A'' of ''X'', either φ is eventually in ''A'' or φ is eventually in ''X''-''A''. PROPERTIES Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of Limit of a Sequence , which is widely used in the theory of Metric Space s. A function ''f'' : ''X'' → ''Y'' between topological spaces is Continuous at the point ''x'' if and only if for every net (''x''α) with :lim ''x''α = ''x'' we have :lim ''f''(''x''α) = ''f''(''x''). Note that this theorem is in general not true if we replace "net" by "sequence". We have to allow for more directed sets than just the natural numbers if ''X'' is not First-countable . In general, a net in a space ''X'' can have more than one limit, but if ''X'' is a Hausdorff Space , the limit of a net, if it exists, is unique. Conversely, if ''X'' is not Hausdorff, then there exists a net on ''X'' with two distinct limits. Thus the uniqueness of the limit is ''equivalent'' to the Hausdorff condition on the space, and indeed this may be taken as the definition. Note that this result depends on the directedness condition; a set indexed by a general preorder or partial order may have distinct limit points even in a Hausdorff space. If ''U'' is a subset of ''X'', then ''x'' is in the Closure of ''U'' if and only if there exists a net (''x''α) with limit ''x'' and such that ''x''α is in ''U'' for all α. In particular, ''U'' is closed if and only if, whenever (''x''α) is a net with elements in ''U'' and limit ''x'', then ''x'' is in ''U''. A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet. A space ''X'' is Compact if and only if every net (''x''α) in ''X'' has a subnet with a limit in ''X''. This can be seen as a generalization of the Bolzano-Weierstrass Theorem and Heine-Borel Theorem . In a Metric Space or Uniform Space , one can speak of '' Cauchy Net s'' in much the same way as Cauchy Sequence s. The concept even generalises to Cauchy Space s. SEE ALSO The theory of Filter s also provides a definition of ''convergence'' in general topological spaces. REFERENCE E. H. Moore and H. L. Smith (1922). A General Theory of Limits. ''American Journal of Mathematics'' 44 (2), 102–121. |
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