Dante space

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