Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a *cover* of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.


If ${\frak U}$ and $\frak{W}$ are collections of covers of a set, we define the property ${\frak U}$ *choose* ${\frak W}$ as follows:



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> > ${\frak U} \choose {\frak W}$: For each ${\cal U}\in {\frak U}$ there is ${\cal W}\subseteq {\cal U}$ such that ${\cal W}\in{\frak W}$.
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We consider the following kinds of open covers of a topological space $X$: An open cover ${\cal U}$ is said to be:


1. a *large* cover if every $x\in X$ is contained in infinitely many members of ${\cal U}$;
2. an $\omega$-cover if every finite subset of $X$ is contained in some member of ${\cal U}$, but $X\notin{\cal U}$;
3. a $\tau$-cover if it is large and for all $x,y\in X$ either $\{U\in {\cal U}: x\in U, y\notin U\}$ is finite or $\{U\in {\cal U}: y\in U, x\notin U\}$ is finite;
4. a $\gamma$-cover if ${\cal U}$ is infinite and every $x\in X$ belongs to all but finitely many members of ${\cal U}$.


Let $\Omega, \text{T}, \Gamma$ denote the collection of $\omega$-, $\tau$- and $\gamma$-covers, respectively.


A topological space $X$ is called a *$D$-space* if if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ covers $X$.


I'm interested in implications (or counterexamples - spaces having one property but not the other) between the 3 properties


* $D$;
* $\Omega \choose \text{T}$;
* $\Omega \choose \Gamma$.


(One implication is trivial: $\Omega \choose \Gamma$ implies $\Omega \choose \text{T}$ because for every space $X$ we have $\Gamma\subseteq \text{T}$.)



Implications between different covering properties of spaces
Aurichi proved that [Every Menger space is D](http://topology.auburn.edu/tp/reprints/v36/tp36009p1.pdf). The last two properties imply Menger,
so they imply D. On the other hand, Menger's property does not imply $\Omega \choose \text{T}$, for example since the latter is equivalent to $S\_{fin}(\Omega,\text{T})$
(Details are available [here](http://arxiv.org/pdf/1404.2239)) and thus implies $S\_{fin}(\Omega,\Omega)$, 
which is the same as Menger's property *in all finite powers*.
In summary, we have the following implications
$${\Omega \choose \Gamma}\Rightarrow{\Omega \choose \text{T}}\Rightarrow D$$
and the last implication is not reversible.


The question whether the first implication is actually an equivalence
is one of the most notorious and important [open problems in the field of selection principles](http://arxiv.org/abs/math/0609601).


