# Net (of sets in a topological space)

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network (of sets in a topological space)

A family of subsets of a topological space such that for each and each neighbourhood of there is an element of such that .

The family of all one-point subsets of a space and every base of a space are networks. The difference between a network and a base is that the elements of a network need not be open sets. Networks appear under continuous mappings: If is a continuous mapping of a topological space onto a topological space and is a base of , then the images of the elements of under form a network in . Further, if is covered by a family of subspaces, then, taking for each any base of and amalgamating these bases, a network in is obtained. Spaces with a countable network are characterized as images of separable metric spaces under continuous mappings.

The minimum cardinality of a network of a space is called the network weight, or net weight, of and is denoted by . The net weight of a space never exceeds its weight (cf. Weight of a topological space), but, as is shown by the example of a countable space without a countable base, the net weight can differ from the weight. For compact Hausdorff spaces the net weight coincides with the weight. This result extends to locally compact spaces, Čech-complete spaces and feathered spaces (cf. Feathered space). Hence, in particular, it follows that weight does not increase under surjective mappings of such spaces. Another corollary: If a feathered space (in particular, a Hausdorff compactum) is given as the union of a family of cardinality of subspaces, the weight of each of which does not exceed , supposed infinite, then the weight of does not exceed .

#### References

 [1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) [2] A.V. Arkhangel'skii, "An addition theorem for weights of sets lying in bicompacta" Dokl. Akad. Nauk SSSR , 126 : 2 (1959) pp. 239–241 (In Russian)