Difference between revisions of "Dante space"
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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]]. | 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]]. | ||
Latest revision as of 21:08, 27 December 2014
2020 Mathematics Subject Classification: Primary: 54A25 [MSN][ZBL]
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 |
Dante space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dante_space&oldid=35902