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For an infinite cardinal $\kappa$, a space $X$ is called $\kappa$-monolithic if $nw(\overline{A}) \le \kappa $ for any set $A \subset X$ with $|A| \le \kappa$. And you can see this definition of monotonically monolithic here

My question is this: What is the relation between $ \kappa$-monolithic and monotonically monolithic? Is the condition of monotonically monolithic stonger than $ \kappa$-monolithic?

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2 Answers 2

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First I'll retype (and expand) your (linked to) definition for future reference, as I don't know how long the link will last:

A collection $\mathcal{N}$ of subsets of $X$ is called an external network of $A \subset X$, when for every $x \in A$ and every neighbourhood $U$ of $x$, there exists some $N \in \mathcal{N}$ such that $x \in N \subset U$. Note that in that case $\left\{ N \cap A: N \in \mathcal{N} \right\}$ is a network for the subspace $A$ (where a network is like a base, except that its members need not be open sets), so it's like a network for $A$ except that they are allowed to "stick out of" $A$. Also note that the monotonic condition wouldn't make sense if we wanted networks of the sets, instead of external networks.

Now, a space $X$ is called monotonically monolithic when for every subset $A$ of $X$ we assign an external network $\mathcal{O}(A)$ of $\overline{A}$, such that the following conditions hold:

1) $\left| \mathcal{O}(A) \right| \le \max\{|A|,\omega\}$ for all $A \subset X$.

2) For all $A \subset B \subset X$, $\mathcal{O}(A) \subset \mathcal{O}(B)$.

3) For all ordinals $\alpha$ and families (of subsets of $X$) $(A_\beta)_{\beta < \alpha}$ such that $\beta < \beta' < \alpha$ implies $A_{\beta} \subset A_{\beta'}$, then $\mathcal{O}(\cup_{\beta < \alpha} A_\beta) = \cup_{\beta < \alpha} \mathcal{O}(A_\beta)$.

Note that $\mathcal{O}(A) \cap \overline{A}$ is a network for $\overline{A}$, and so $\operatorname{nw}(\overline{A}) \le |\mathcal{O}(A)|$, so condition (1) shows that $X$ is monolithic as witnessed by the sets $\mathcal{O}(A) \cap \overline{A}$. So it's clear that this property is at least as strong as being monolithic. Condition (2) is the monotonicity condition, quite similar e.g. to how normality is related to monotonic normality, which is one of the first properties that was "monotonized" in this way (and it turns out to be quite a bit stronger than normality). Condition (3) is more technical, and is a sort of continuity condition (note that the right into left inclusion already follows from monotonicity).

Now, it is well-known that all Corson compact spaces (see here) are monolithic, and Gruenhage in example 2.3 of his paper has an example of a Corson compact space that is not $\omega$-monotonically monolithic (which is just the above definition restricted to only countable subsets $A$ of $X$). So this notion is strictly stronger.

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The answer to your question is that monotone monolithicity is stronger than $\kappa$-monolithicity.

In

O.T. Alas & V.V. Tkachuk & R.G. Wilson, A broader context for monotonically monolithic spaces, Acta Math. Hungar. 125, pp.369–385, MR2564435

the authors present a definition for monotonically $\kappa$-monolithic spaces which is stronger than $\kappa$-monolithicity, and such that monotone monolithicity is simply monotone $\kappa$-monolithicity for all infinite $\kappa$ (just as monolithicity is $\kappa$-monolithicity for all infinite $\kappa$).

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