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A type of topological space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301201.png" /> be a topological space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301202.png" /> be a subspace of it and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301204.png" /> be infinite cardinals. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301205.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301207.png" />-monolithic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301208.png" /> if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d0301209.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012010.png" /> the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012012.png" /> is a compactum of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012013.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012014.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012016.png" />-suppresses the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012017.png" /> if it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012020.png" /> that there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012021.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012023.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012024.png" /> is said to be a Dante space if for each infinite cardinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012025.png" /> there exists an everywhere-dense subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012027.png" /> which is both monolithic in itself and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012028.png" />-suppressed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030120/d03012029.png" />. The class of Dante spaces contains the class of dyadic compacta (cf. [[Dyadic compactum|Dyadic compactum]]).
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A type of [[topological space]]. Let $X$ be a topological space, let $Y$ be a subspace of $X$ and let $\tau$ and $\lambda$ be infinite [[cardinal]]s. The space $Y$ is said to be $\tau$-monolithic in $X$ if for each $A \subseteq Y$ such that $\mathrm{card}(A) \le \tau$ the closure $[A]$ in $X$ is a compactum of [[Weight of a topological space|weight]] $\le \tau$. The space $X$ $\tau$-suppresses the subspace $Y$ if it follows from $\lambda \ge \tau$, $A \subseteq Y$ and $\mathrm{card}(A) \le \exp \tau$ that there exists an $A' \subseteq X$ for which $[A'] \supseteq A$ and $\mathrm{card}(A') \le \lambda$. The space $X$ is said to be a Dante space if for each infinite cardinal $\tau$ there exists an [[ Everywhere-dense set|everywhere-dense subspace]] $Y$ in $X$ which is both monolithic in itself and is $\tau$-suppressed by $X$. The class of Dante spaces contains the class of [[Dyadic compactum|dyadic compacta]].
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  "Factorization theorems and spaces of continuous functions: stability and monolithicity"  ''Sov. Math. Dokl.'' , '''26'''  (1982)  pp. 177–181  ''Dokl. Akad. Nauk SSSR'' , '''265''' :  5  (1982)  pp. 1039–1043</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  "Factorization theorems and spaces of continuous functions: stability and monolithicity"  ''Sov. Math. Dokl.'' , '''26'''  (1982)  pp. 177–181  ''Dokl. Akad. Nauk SSSR'' , '''265''' :  5  (1982)  pp. 1039–1043</TD></TR>
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</table>

Revision as of 21:06, 27 December 2014


A type of topological space. Let $X$ be a topological space, let $Y$ be a subspace of $X$ and let $\tau$ and $\lambda$ be infinite cardinals. The space $Y$ is said to be $\tau$-monolithic in $X$ if for each $A \subseteq Y$ such that $\mathrm{card}(A) \le \tau$ the closure $[A]$ in $X$ is a compactum of weight $\le \tau$. The space $X$ $\tau$-suppresses the subspace $Y$ if it follows from $\lambda \ge \tau$, $A \subseteq Y$ and $\mathrm{card}(A) \le \exp \tau$ that there exists an $A' \subseteq X$ for which $[A'] \supseteq A$ and $\mathrm{card}(A') \le \lambda$. The space $X$ is said to be a Dante space if for each infinite cardinal $\tau$ there exists an everywhere-dense subspace $Y$ in $X$ which is both monolithic in itself and is $\tau$-suppressed by $X$. The class of Dante spaces contains the class of dyadic compacta.


Comments

For applications of these notions see [a1].

References

[a1] A.V. Arkhangel'skii, "Factorization theorems and spaces of continuous functions: stability and monolithicity" Sov. Math. Dokl. , 26 (1982) pp. 177–181 Dokl. Akad. Nauk SSSR , 265 : 5 (1982) pp. 1039–1043
How to Cite This Entry:
Dante space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dante_space&oldid=16209
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article