View source for Net (of sets in a topological space)

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''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|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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>

====Comments====
Most English-language texts (cf. e.g. [[#References|[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|directed set]]. In Russian this is called a [[Generalized sequence|generalized sequence]].

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

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