Filter (mathematics) - Filter On A Set - Filters in Topology

Filters in Topology

In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space.

In topology and related areas of mathematics, a filter is a generalization of a net. Both nets and filters provide very general contexts to unify the various notions of limit to arbitrary topological spaces.

A sequence is usually indexed by the natural numbers, which are a totally ordered set. Thus, limits in first-countable spaces can be described by sequences. However, if the space is not first-countable, nets or filters must be used. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. Filters can be thought of as sets built from multiple nets. Therefore, both the limit of a filter and the limit of a net are conceptually the same as the limit of a sequence.

Other articles related to "filters, filter, filters in topology":

Camfil - Segments - Comfort Air
... Most air filters for this segment are installed in ventilation systems in schools, offices, homes, hospitals and airports, among other facilities ... Air market consists mainly of replacement filters, operations in this segment are relatively insensitive to fluctuations in the economy ... In Europe, bag filters are dominant, with Hi-Flo being the most common product ...
Constant K Filter
... Constant k filters, also k-type filters, are a type of electronic filter designed using the image method ... They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components ... Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of ...
Filter (mathematics) - Filter On A Set - Filters in Topology - Cauchy Filters
... To say that a filter base B on X is Cauchy means that for each real number ε>0, there is a B0 ∈ B such that the metric diameter of B0 is less than ε ... (xn) is a Cauchy sequence if and only if the filter base {{xN,xN+1...} N ∈ {1,2,3...} } is Cauchy ... More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an A ∈ F with (x,y) ∈ U for all x,y ∈ A ...
Camfil - Segments - Clean Processes
... specifically for the biopharmaceutical industry, including high-temperature filters and bag-in, bag-out systems for containing contaminants ... stages, where pre-filtration is first carried out to protect the high-efficiency filters, which are much more complex and consequently more expensive ... Pre-filters here are similar to those used in the Comfort Air sector ...
Constant K Filter - History
... Constant k filters were invented by George Campbell ... He published his work in 1922, but had clearly invented the filters some time before, as his colleague at AT T Co, Otto Zobel, was already making improvements to the design at this time ... Campbell's filters were far superior to the simpler single element circuits that had been used previously ...

Famous quotes containing the word filters:

Raise a million filters and the rain will not be clean, until the longing for it be refined in deep confession. And still we hear, If only this nation had a soul, or, Let us change the way we trade, or, Let us be proud of our region.
Leonard Cohen (b. 1934)