What classes of spaces $X$ have the property that that for every countable subset $C \subset X$, $\overline{C}$ does not have an uncountable closed discrete subset?
I know every space with countable extent, such as compact spaces, Lindelöf spaces and so on, shall satisfy this property. Are there other examples with this property?
It seems the following.
I shall call such spaces good, as a working term.
Counterexamples. One of the simplest examples of not a good space is the square of the Sorgenfrey arrow, that is the real line endowed with the topology generated by the base $\{[a,b):a<b\}$. It is a separable space containig a closed discrete set $\{(x,-x) : x\in\Bbb R\}$ of size $\frak c$. An other example is Mrówka space. Also there is a separable Moore space $X$ with uncountable extent (see [DRRT, 3.2.3.1]). In my PhD thesis I constructed a refinement $\tau’$ of the standard topology of the space $\Bbb R$ which is a Tychonoff first countable Baire quasitopological group with a countable $\pi$-base (and, therefore, separable) $e(\Bbb R,\tau)=\frak c$ [Rav1, Example 5.11], [Rav2, 5.20] by putting the base of the topology $\tau'$ a family ${\mathcal B}(x)=\{U_n(x):n\in\Bbb N\}$ at every point $x\in\Bbb R$, where $U_n(x)=x+(\pm 1+(-1/2;1/2)n^{-1})\cdot\{6^{-m}:m\ge n\}$. (More properties of this space are described in my PhD Thesis).
Main positive result. If each open cover of cardinality $\omega_1$ of a space $X$ has a point countable open refinement then the space $X$ is good (in particular, each meta-Lindelöf space is good) . Indeed, assume that there exists a countable subset $C$ of the space $X$ and an uncountable closed discrete set $D’\subset\overline{C}$. Let $D\subset D’$ be an arbitrary set of cardinality $\omega_1$. Since the set $D’$ is closed and discrete subset of the space $X$, the set $D$ is closed and discrete subset of the space $X$ too. Since the set $D$ is a discrete subset of the space $X$, each point $x\in D$ has an open neighborhood $U_x\subset X$ such that $U_x\cap D=\{x\}$. Let $\mathcal U=\{X\setminus D\}\cup\{U_x:x\in D\}$ be an open cover of the space $X$. Then there exists a point countable open refinement $\mathcal V$ of the cover $\mathcal U$. Since for each member $V\in\mathcal V$ $|V\cap X|\le 1$, for each point $c\in C$ a set $S_c=\{x\in D:$ there exists a member $V\in\mathcal V$ such that both points $c$ and $x$ belong to $V\}$ is countable. Since $D\subset\overline{C}$, $D\subset\bigcup \{S_c: c\in C\}$. So the set $D$ is countable, a contradiction.
An other class of good spaces is comprised by spaces in which all countable subsets are closed (for instance, $T_1$ P-spaces).
Operations. The Tychonoff counterexamples shows that a subspace of a good space (Tychonoff cube) need not to be good. But a closed subspace of a good space is good. The Sorgenfrey arrow is a Lindelöf (and, hence, a good) space, which square is not good. Since each space is a continuous image of a discrete (and, therefore, of a good) space, good spaces are not preserved by continuous maps. I did not investigate closed, open or quotient images of good spaces.
Normality. Clearly, each collectively normal (and even collectively Hausdorff) space is good. I recall that a topological space $X$ is called collectively normal (resp. collectively Hausdorff if for each discrete family $\mathcal F $ of (finite) subsets of $X$ there is a discrete family $(U_F)_{F\in\mathcal F }$ of open sets such that $F\subset U_F$ for all $F\in\mathcal F $.
I don’t know, is each perfectly normal space good.
Jones showed that under $\frak c < 2^{\omega_1}$ a normal separable spaces cannot contain an uncountable closed discrete subset, that is all normal spaces are good. (I think it can be proved similarly to Example 1.5.10 from [Eng], showing that a separable normal space cannot have a closed discrete subset of cardinality $\frak c$). Nevertheless, from the other side, there are consistent examples of separable normal spaces $X$ such that $e(X)=\mathfrak c$, see [LM]. Thus, the goodness of normal spaces is independent in ZFC.
Countable covers. I shall follow a paper [MS] and use definitions from it.
Lemma. The following conditions are equivalent:
Each countably paracompact space is good.
There is no separable countably paracompact space with an uncountable closed discrete subset.
There is no separable countably semi-paracompact space with an uncountable closed discrete subset.
There is no separable space $X$ with an uncountable closed discrete suset of the space $X$ which is relatively countably semi-paracompact.
Proof. The implications $(4)\Rightarrow (3)$ and $(3)\Rightarrow (2)$ are obvious.
$(2)\Rightarrow (1)$. Assume that $X$ is not a good countably paracompact space. Then $X$ has a separable closed subspace $Y$ containing an uncountable closed discrete subset. Since $Y$ is a closed subspace of a countably paracompact space, $Y$ is a countably paracompact space too, which contradicts to (1).
$(1)\Rightarrow (4)$. Assume not (4), that is there is a separable space $X$ with uncountable closed discrete subspace of the space $X$ which is relatively countably semi-paracompact. Then by Corollary 3.10, there exists a small dominating family. Watson [Wat] proved that the existence of a small dominating family is equivalent to the existence of a separable countably paracompact space with an uncountable closed discrete subset. This implies a contradiction with (1).$\square$
By Corollary 3.10, conditions of Lemma are satisfied under $cf(\frak c)=\frak c<2^{\omega_1}$ $+$ there is no inner model with a measurable cardinal. The question: “can these assumptions be weakened to $\frak c<2^{\omega_1}$?” remains open. [MS, 5.2]
By Proposition 4.1 there is a countably metacompact, orthocompact, separable space with a closed discrete subset of size $\frak c$ (and, hence, not a good space) and it is consistent with ZFC $+$ $\frak c<2^{\omega_1}$ that there is a countably metacompact, orthocompact, pseudonormal, separable space with uncountable extent (and, hence, not a good space too).
There are models of $\frak c=2^{\omega_1}$ in which it is easy to find examples of separable, normal, countably paracompact (a)-spaces for which extent is not constrained by density. It suffices to get a model of $\omega_1 <\frak p$ and, taking a subset $Y$ of $\Bbb R$ of size $\omega_1$ in this model, consider the well-known space $M(Y)$, the Moore space derived from $Y$. If $|Y| =\omega_1 <\frak p$, then $\frak c=2^{\omega_1}$ and $M(Y)$ is a separable, normal, countably paracompact space with uncountable extent (see [20] for more details [In fact, if $|Y| <\frak p$ then $M(Y)$ also satisfies a related topological property: Matveev’s property (a). In previous works of the authors, this property was extensively studied, side-by-side with normality and countable paracompactness (see [14], [15], [16], [20] and [22] for results and references).]). Therefore, it is reasonable to suppose that the general problem of finding topological and set-theoretical conditions implying $e(X)\le d(X)$ should be investigated in models of $\frak c=2^{\omega_1}$.
Authors finish the paper as follows:
Despite the obstructions presented in Section 4, the authors are still very interested in investigating the borders between countable paracompactness and countable metacompactness – specifically, with regard to the general problem under consideration: when does density constrain extent?
Problem 5.3. Find topological properties $\mathcal P$ such that countably metacompact spaces which satisfy $\mathcal P$ have their extents constrained in terms of their densities. Obviously, we are interested in properties $\mathcal P$ which neither imply countable paracompactness when combined with countable metacompactness, nor are satisfied by every Isbell-Mrówka space.
References
[DRRT] E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree, Star covering properties. Topology Appl, 39:1 (1991), 71-103.
[Eng] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
[LM] R. Levy, M. Matveev, Weak extent in normal spaces, Comment. Math. Univ. Carolin. 46:3 (2005), 497-501.
[MS] C. J. G. Morgan, S. G. da Silva, Constraining extent by density: on generalizations of normality and countable paracompactness, Bol. Soc. Mat. Mexicana (3) Vol. 18, 2012.
[Rav1] A. V. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis. - Lviv University, 2002, (in Ukrainian).
[Rav2] A. V. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis. - Lviv University, 2002, (draft and unchecked English version). (the link will be added later)
[Wat] W. S. Watson, Separation in countably paracompact spaces, Trans. Amer. Math. Soc., 290, (1985), 831-842.
