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A simple example of a filter is the Fréchet filter on an infinite set S. This is the set of all subsets of S which have finite complement. |
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A simple example of a filter is the set of all subsets of S which contain a common point x. The Fréchet filter on an infinite set S is the set of all subsets of S which have finite complement. |
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Filters are useful in topology. For example, the set of all neighbourhoods of a point x in a topological space is a filter, called the neighbourhood filter of x. A filter which is a superset of the neighbourhood filter of x is said to converge to x. (Note that in a non-Hausdorff space a filter can converge to more than one point.) |
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Filters are useful in topology: they play the role of sequences in metric spaces. The set of all neighbourhoods of a point x in a topological space is a filter, called the neighbourhood filter of x. A filter which is a superset of the neighbourhood filter of x is said to converge to x. Note that in a non-Hausdorff space a filter can converge to more than one point. |
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