Net (of sets in a topological space)

network (of sets in a topological space)

A family $ {\mathcal P} $ of subsets of a topological space $ X $ such that for each $ x \in X $ and each neighbourhood $ O _ {x} $ of $ x $ there is an element $ M $ of $ {\mathcal P} $ such that $ x \in M \subset O _ {x} $.

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 $ f $ is a continuous mapping of a topological space $ X $ onto a topological space $ Y $ and $ {\mathcal B} $ is a base of $ X $, then the images of the elements of $ {\mathcal B} $ under $ f $ form a network $ {\mathcal P} = \{ {f U } : {U \in {\mathcal B} } \} $ in $ Y $. Further, if $ X $ is covered by a family $ \{ {X _ \alpha } : {\alpha \in A } \} $ of subspaces, then, taking for each $ \alpha \in A $ any base $ {\mathcal B} _ \alpha $ of $ X _ \alpha $ and amalgamating these bases, a network $ {\mathcal P} = \cup \{ { {\mathcal B} _ \alpha } : {\alpha \in A } \} $ in $ X $ 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 $ X $ is called the network weight, or net weight, of $ X $ and is denoted by $ \mathop{\rm nw} ( X) $. 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 $ X $( in particular, a Hausdorff compactum) is given as the union of a family of cardinality $ \leq \tau $ of subspaces, the weight of each of which does not exceed $ \tau $, supposed infinite, then the weight of $ X $ does not exceed $ \tau $.

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)

Comments

Most English-language texts (cf. e.g. [a4]) use network for the concept called "net" above. This is because the term "net" also has a second, totally different, meaning in general topology.

A net in a set (topological space) $ X $ is an indexed set $ \{ x _ \alpha \} _ {\alpha \in \Sigma } $ of points of $ X $, where $ \Sigma $ is a directed set. In Russian this is called a generalized sequence.

One can build a theory of convergence for nets: Moore–Smith convergence (cf. Moore space).

References

[a1]	R. Engelking, "General topology" , PWN (1977) (Translated from Polish) (Revised and extended version of [3] above)
[a2]	J.L. Kelley, "Convergence in topology" Duke Math. J. , 17 (1950) pp. 277–283
[a3]	E.H. Moore, H.L. Smith, "A general theory of limits" Amer. J. Math. , 44 (1922) pp. 102–121
[a4]	J.-I. Nagata, "Modern general topology" , North-Holland (1985)