Recall that a net is a function from a directed set to a set . The value of at is usually denoted by . Let be a subset of . We say that a net is frequently in if for every , there is a such that and .
Suppose a net is frequently in . Let . Then is a cofinal subset of , for if , then by definition of , there is such that , and therefore .
The notion of “frequently in” is related to the notion of “eventually in” in the following sense: a net is eventually in a set iff it is not frequently in , its complement. Suppose is eventually in . There is such that for all , or equivalently, for no . The converse is can be argued by tracing the previous statements backwards.
In a topological space , a point is said to be a cluster point of a net (or, occasionally, clusters at ) if is frequently in every neighborhood of . In this general definition, a limit point is always a cluster point. But a cluster point need not be a limit point. As an example, take the sequence has as a cluster point. But clearly is not a limit point, as the sequence diverges in .
|Date of creation||2013-03-22 17:14:23|
|Last modified on||2013-03-22 17:14:23|
|Last modified by||CWoo (3771)|
|Defines||cluster point of a net|