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Difference between revisions of "Weakly infinite-dimensional space"

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A [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972601.png" /> such that for any infinite system of pairs of closed subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972602.png" /> of it,
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A [[topological space]] $X$ such that for any infinite system of pairs of closed subsets $(A_i,B_i)$ of it,
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972603.png" /></td> </tr></table>
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A_i \cap b_i = \emptyset,\ \ i=1,2,\ldots
 
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$$
there are partitions (cf. [[Partition|Partition]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972604.png" /> (between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972606.png" />) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972607.png" />. An infinite-dimensional space which is not weakly infinite dimensional is called strongly infinite dimensional. Weakly infinite-dimensional spaces are also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w0972609.png" />-weakly infinite dimensional. If in the above definition it is further required that some finite subfamily of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w09726010.png" />'s have empty intersection, one obtains the concept of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w09726012.png" />-weakly infinite-dimensional space.
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there are [[partition]]s $C_i$ (between $A_i$ and $B_i$) such that $\cap C_i = \emptyset$. An infinite-dimensional space which is not weakly infinite dimensional is called strongly infinite dimensional. Weakly infinite-dimensional spaces are also called $A$-weakly infinite dimensional. If in the above definition it is further required that some finite subfamily of the $C_i$ have empty intersection, one obtains the concept of an $S$-weakly infinite-dimensional space.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  B.A. Pasynkov,  "Introduction to dimension theory" , Moscow  (1973)  (In Russian)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
In addition to the above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w09726013.png" />-weakly stands for Aleksandrov weakly, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w09726014.png" />-weakly for Smirnov weakly. There is also the obsolete notion of Hurewicz-weakly infinite-dimensional space. Cf. the survey [[#References|[a1]]].
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In addition to the above, $A$-weakly stands for Aleksandrov weakly, and $S$-weakly for Smirnov weakly. There is also the obsolete notion of Hurewicz-weakly infinite-dimensional space. Cf. the survey [[#References|[a1]]].
  
To avoid ambiguity in the phrase  "infinite-dimensional space" , the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097260/w09726015.png" /> could be required to be metrizable, cf. [[#References|[a2]]].
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To avoid ambiguity in the phrase  "infinite-dimensional space" , the space $X$ could be required to be metrizable, cf. [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Some results in the theory of topological spaces, obtained within the last twenty-five years"  ''Russian Math. Surveys'' , '''15''' :  2  (1960)  pp. 23–83  ''Uspekhi Mat. Nauk'' , '''15''' :  2  (1960)  pp. 25–95</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. van Mill,  "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1989)  pp. 40</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Engelking,  E. Pol,  "Countable-dimensional spaces: a survey"  ''Diss. Math.'' , '''216'''  (1983)  pp. 5–41</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Some results in the theory of topological spaces, obtained within the last twenty-five years"  ''Russian Math. Surveys'' , '''15''' :  2  (1960)  pp. 23–83  ''Uspekhi Mat. Nauk'' , '''15''' :  2  (1960)  pp. 25–95</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. van Mill,  "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1989)  pp. 40</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Engelking,  E. Pol,  "Countable-dimensional spaces: a survey"  ''Diss. Math.'' , '''216'''  (1983)  pp. 5–41</TD></TR>
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</table>
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Revision as of 19:01, 4 October 2017

A topological space $X$ such that for any infinite system of pairs of closed subsets $(A_i,B_i)$ of it, $$ A_i \cap b_i = \emptyset,\ \ i=1,2,\ldots $$ there are partitions $C_i$ (between $A_i$ and $B_i$) such that $\cap C_i = \emptyset$. An infinite-dimensional space which is not weakly infinite dimensional is called strongly infinite dimensional. Weakly infinite-dimensional spaces are also called $A$-weakly infinite dimensional. If in the above definition it is further required that some finite subfamily of the $C_i$ have empty intersection, one obtains the concept of an $S$-weakly infinite-dimensional space.

References

[1] P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian)


Comments

In addition to the above, $A$-weakly stands for Aleksandrov weakly, and $S$-weakly for Smirnov weakly. There is also the obsolete notion of Hurewicz-weakly infinite-dimensional space. Cf. the survey [a1].

To avoid ambiguity in the phrase "infinite-dimensional space" , the space $X$ could be required to be metrizable, cf. [a2].

References

[a1] P.S. Aleksandrov, "Some results in the theory of topological spaces, obtained within the last twenty-five years" Russian Math. Surveys , 15 : 2 (1960) pp. 23–83 Uspekhi Mat. Nauk , 15 : 2 (1960) pp. 25–95
[a2] J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1989) pp. 40
[a3] R. Engelking, E. Pol, "Countable-dimensional spaces: a survey" Diss. Math. , 216 (1983) pp. 5–41
How to Cite This Entry:
Weakly infinite-dimensional space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weakly_infinite-dimensional_space&oldid=12749
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article