Implications between different covering properties of spaces

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:

${\frak U} \choose {\frak W}$: For each ${\cal U}\in {\frak U}$ there is ${\cal W}\subseteq {\cal U}$ such that ${\cal W}\in{\frak W}$.

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 $\Gamma\subseteq \text{T}$.)

• $\{X\}$ is always an $\omega$-cover, so no space is either $\Omega \choose T$ or $\Omega \choose \Gamma$. – Ramiro de la Vega Jul 29 '15 at 18:30
• Oops - I have to exclude this "pathological" cover -- edited the post accordingly. Thanks for noticing – Dominic van der Zypen Jul 30 '15 at 6:35

Aurichi proved that Every Menger space is D. 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) 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.