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

''network (of sets in a topological space)''

A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663601.png" /> of subsets of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663602.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663603.png" /> and each neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663604.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663605.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663607.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663608.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n0663609.png" /> is a continuous mapping of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636010.png" /> onto a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636012.png" /> is a base of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636013.png" />, then the images of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636014.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636015.png" /> form a network <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636017.png" />. Further, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636018.png" /> is covered by a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636019.png" /> of subspaces, then, taking for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636020.png" /> any base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636022.png" /> and amalgamating these bases, a network <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636025.png" /> is called the network weight, or net weight, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636026.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636027.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636028.png" /> (in particular, a Hausdorff compactum) is given as the union of a family of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636029.png" /> of subspaces, the weight of each of which does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636030.png" />, supposed infinite, then the weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636031.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636032.png" />.

====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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636033.png" /> is an indexed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636034.png" /> of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066360/n06636036.png" /> 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>