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''network (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" />.
+
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 <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 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 <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" />.
+
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====
 
====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>
 
<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====
 
====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.
 
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]].
+
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]]).
 
One can build a theory of convergence for nets: Moore–Smith convergence (cf. [[Moore space|Moore space]]).

Latest revision as of 08:02, 6 June 2020


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)
How to Cite This Entry:
Net (of sets in a topological space). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Net_(of_sets_in_a_topological_space)&oldid=17332
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article