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''of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200801.png" />''
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A [[Cardinal number|cardinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200802.png" /> such that every family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200803.png" /> of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200804.png" />, consisting of non-empty open subsets of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200805.png" />, contains a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200806.png" />, also of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200807.png" />, with non-empty intersection, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200808.png" />. A regular uncountable cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c0200809.png" /> is a calibre of a topological product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008011.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008012.png" /> is a calibre of every factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008013.png" />. The property of being a calibre is preserved under continuous mappings; every uncountable regular cardinal number is a calibre of any dyadic compactum. If the first uncountable cardinal number is a calibre of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008015.png" /> satisfies the [[Suslin condition|Suslin condition]]. In some models of set theory the converse is almost true, namely, Martin's axiom and the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008016.png" /> imply the following: If a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008017.png" /> satisfies the Suslin condition, then every uncountable family of non-empty open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008018.png" /> contains an uncountable centred subfamily. In particular, in this model, the cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008019.png" /> is a calibre for every compactum with the Suslin condition. In some other models of set theory, a compactum with the Suslin condition exists for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008020.png" /> is not a calibre.
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''of a topological space  $  X $''
 +
 
 +
A [[Cardinal number|cardinal number]] $  \tau $
 +
such that every family $  \mathfrak B $
 +
of cardinality $  \tau $,  
 +
consisting of non-empty open subsets of a topological space $  X $,  
 +
contains a subset $  \mathfrak B  ^  \prime  \subset  \mathfrak B $,  
 +
also of cardinality $  \tau $,  
 +
with non-empty intersection, i.e. $  \cap \{ {U } : {U \in \mathfrak B  ^  \prime  } \} \neq \emptyset $.  
 +
A regular uncountable cardinal number $  \tau $
 +
is a calibre of a topological product $  \prod X _  \alpha  $,  
 +
$  \alpha \in A $,  
 +
if and only if $  \tau $
 +
is a calibre of every factor $  X _  \alpha  $.  
 +
The property of being a calibre is preserved under continuous mappings; every uncountable regular cardinal number is a calibre of any dyadic compactum. If the first uncountable cardinal number is a calibre of a space $  X $,  
 +
then $  X $
 +
satisfies the [[Suslin condition|Suslin condition]]. In some models of set theory the converse is almost true, namely, Martin's axiom and the condition $  \aleph _ {1} < 2 ^ {\aleph _ {0} } $
 +
imply the following: If a space $  X $
 +
satisfies the Suslin condition, then every uncountable family of non-empty open sets in $  X $
 +
contains an uncountable centred subfamily. In particular, in this model, the cardinal number $  \aleph _ {1} $
 +
is a calibre for every compactum with the Suslin condition. In some other models of set theory, a compactum with the Suslin condition exists for which $  \aleph _ {1} $
 +
is not a calibre.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.A. Suslin,  "On the product of topological spaces"  ''Trudy. Mat. Inst. Steklov'' , '''24'''  (1948)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.A. Suslin,  "On the product of topological spaces"  ''Trudy. Mat. Inst. Steklov'' , '''24'''  (1948)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The spelling caliber is more common.
 
The spelling caliber is more common.
  
Usually, calibers are defined using indexed collections of open sets. In that case a [[Cardinal number|cardinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008021.png" /> is a caliber of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008022.png" /> if and only if for every collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008023.png" /> of non-empty open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008024.png" /> there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008025.png" /> of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008027.png" />.
+
Usually, calibers are defined using indexed collections of open sets. In that case a [[Cardinal number|cardinal number]] $  \kappa $
 +
is a caliber of $  X $
 +
if and only if for every collection $  \{ {U _  \alpha  } : {\alpha \in \kappa } \} $
 +
of non-empty open subsets of $  X $
 +
there is a set $  A \subset  \kappa $
 +
of size $  \kappa $
 +
such that $  \cap _ {\alpha \in A }  U _  \alpha  \neq \emptyset $.
  
One also considers precalibers: a cardinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008028.png" /> is a precaliber of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008029.png" /> if and only if for every collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008030.png" /> of non-empty subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008031.png" /> there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008032.png" /> of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008034.png" /> has the finite intersection property (i.e. the intersection of any finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008035.png" /> is non-empty). Thus, Martin's axiom (cf. [[Suslin hypothesis|Suslin hypothesis]]) plus the negation of the [[Continuum hypothesis|continuum hypothesis]] imply that every space satisfying the Suslin condition has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008036.png" /> as a precaliber, while for a [[Compact space|compact space]] its calibers and precalibers are the same.
+
One also considers precalibers: a cardinal number $  \kappa $
 +
is a precaliber of $  X $
 +
if and only if for every collection $  \{ {U _  \alpha  } : {\alpha \in \kappa } \} $
 +
of non-empty subsets of $  X $
 +
there is a set $  A \subset  \kappa $
 +
of size $  \kappa $
 +
such that $  \{ {U _  \alpha  } : {\alpha \in \kappa } \} $
 +
has the finite intersection property (i.e. the intersection of any finite number of $  U _  \alpha  $
 +
is non-empty). Thus, Martin's axiom (cf. [[Suslin hypothesis|Suslin hypothesis]]) plus the negation of the [[Continuum hypothesis|continuum hypothesis]] imply that every space satisfying the Suslin condition has $  \aleph _ {1} $
 +
as a precaliber, while for a [[Compact space|compact space]] its calibers and precalibers are the same.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Argyros,  A. Tsarpalias,  "Calibers of compact spaces"  ''Trans. Amer. Math. Soc.'' , '''270'''  (1982)  pp. 149–162</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Broverman,  J. Ginsburg,  K. Kunen,  F.D. Tall,  "Topologies determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008037.png" />-ideals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020080/c02008038.png" />"  ''Canad. J. Math.'' , '''30'''  (1978)  pp. 1306–1312</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.W. Comfort,  S. Negrepontis,  "Chain conditions in topology" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Juhász,  "Cardinal functions. Ten years later" , ''MC Tracts'' , '''123''' , Math. Centre  (1980)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Argyros,  A. Tsarpalias,  "Calibers of compact spaces"  ''Trans. Amer. Math. Soc.'' , '''270'''  (1982)  pp. 149–162</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Broverman,  J. Ginsburg,  K. Kunen,  F.D. Tall,  "Topologies determined by $\sigma$-ideals on $\omega_1$"  ''Canad. J. Math.'' , '''30'''  (1978)  pp. 1306–1312</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  W.W. Comfort,  S. Negrepontis,  "Chain conditions in topology" , Cambridge Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Juhász,  "Cardinal functions. Ten years later" , ''MC Tracts'' , '''123''' , Math. Centre  (1980)</TD></TR></table>

Latest revision as of 09:08, 26 March 2023


of a topological space $ X $

A cardinal number $ \tau $ such that every family $ \mathfrak B $ of cardinality $ \tau $, consisting of non-empty open subsets of a topological space $ X $, contains a subset $ \mathfrak B ^ \prime \subset \mathfrak B $, also of cardinality $ \tau $, with non-empty intersection, i.e. $ \cap \{ {U } : {U \in \mathfrak B ^ \prime } \} \neq \emptyset $. A regular uncountable cardinal number $ \tau $ is a calibre of a topological product $ \prod X _ \alpha $, $ \alpha \in A $, if and only if $ \tau $ is a calibre of every factor $ X _ \alpha $. The property of being a calibre is preserved under continuous mappings; every uncountable regular cardinal number is a calibre of any dyadic compactum. If the first uncountable cardinal number is a calibre of a space $ X $, then $ X $ satisfies the Suslin condition. In some models of set theory the converse is almost true, namely, Martin's axiom and the condition $ \aleph _ {1} < 2 ^ {\aleph _ {0} } $ imply the following: If a space $ X $ satisfies the Suslin condition, then every uncountable family of non-empty open sets in $ X $ contains an uncountable centred subfamily. In particular, in this model, the cardinal number $ \aleph _ {1} $ is a calibre for every compactum with the Suslin condition. In some other models of set theory, a compactum with the Suslin condition exists for which $ \aleph _ {1} $ is not a calibre.

References

[1] N.A. Suslin, "On the product of topological spaces" Trudy. Mat. Inst. Steklov , 24 (1948) (In Russian)

Comments

The spelling caliber is more common.

Usually, calibers are defined using indexed collections of open sets. In that case a cardinal number $ \kappa $ is a caliber of $ X $ if and only if for every collection $ \{ {U _ \alpha } : {\alpha \in \kappa } \} $ of non-empty open subsets of $ X $ there is a set $ A \subset \kappa $ of size $ \kappa $ such that $ \cap _ {\alpha \in A } U _ \alpha \neq \emptyset $.

One also considers precalibers: a cardinal number $ \kappa $ is a precaliber of $ X $ if and only if for every collection $ \{ {U _ \alpha } : {\alpha \in \kappa } \} $ of non-empty subsets of $ X $ there is a set $ A \subset \kappa $ of size $ \kappa $ such that $ \{ {U _ \alpha } : {\alpha \in \kappa } \} $ has the finite intersection property (i.e. the intersection of any finite number of $ U _ \alpha $ is non-empty). Thus, Martin's axiom (cf. Suslin hypothesis) plus the negation of the continuum hypothesis imply that every space satisfying the Suslin condition has $ \aleph _ {1} $ as a precaliber, while for a compact space its calibers and precalibers are the same.

References

[a1] S. Argyros, A. Tsarpalias, "Calibers of compact spaces" Trans. Amer. Math. Soc. , 270 (1982) pp. 149–162
[a2] S. Broverman, J. Ginsburg, K. Kunen, F.D. Tall, "Topologies determined by $\sigma$-ideals on $\omega_1$" Canad. J. Math. , 30 (1978) pp. 1306–1312
[a3] W.W. Comfort, S. Negrepontis, "Chain conditions in topology" , Cambridge Univ. Press (1982)
[a4] I. Juhász, "Cardinal functions. Ten years later" , MC Tracts , 123 , Math. Centre (1980)
How to Cite This Entry:
Calibre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calibre&oldid=17075
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article