Does a space with peoperty A have a topological name? As we know, 

If $X$ is a Tychonoff pseudocompact space, then for every decreasing sequence $\cdots\subset W_2\subset W_1$ of nonempty open subsets of $X$ the intersection $\bigcap_{i=1}^{\infty} \overline{W_i}$ is nonempty.

In this result, the sequence $\{W_i\}$ is countable. If we change the cardinality of the sequence 
to be $\omega_1$, i.e., 
Property A: For every decreasing sequence $\cdots\subset W_\alpha\subset \cdots\subset W_2\subset W_1$ of nonempty open subsets of $X$ the intersection $\bigcap_{\alpha=1}^{\omega_1} \overline{W_i}$ is nonempty.
Does a space with peoperty A have a topological name?
Thanks for your help.
 A: It seems the following.
A space $X$ is almost $\omega_1$-Lindelöf [Par][Mat, p. 92], if every open cover of cardinality at most $\omega_1$ has a countable subfamily whose union is dense in $X$. It is easy to check that a space $X$ has Property A iff $X$ is almost $\omega_1$-Lindelöf. See the diagrams at [Mat, p.92] and at [DRRT, p.94] about the relations of Property A of a space with other similar properties. It seems that  Property (*) implies Property A even without $T_3$ assumption. Property A implies feebly Lindelöfness (Indeed, assume that a space $X$ is not feebly Lindelöf.  Then it  has a locally finite family $\{Y_\alpha: \alpha<\omega_1\}$ of nonempty open sets. For each $\alpha<\omega_1$ put $W_\alpha=\bigcup_{\beta\ge\alpha} Y_\beta$. Then $\{W_\alpha\}$ is a decreasing sequence of nonempty open subsets of the space $X$ such that  $\bigcap \overline{W_i}=\varnothing$). Feebly Lindelöfness implies  DCCC [Mat, p.87-88]. Example 22 at [Mat, p.88] is Hausdorff DCCC, but not feebly Lindelöf. But each regular DCCC space is feeble Lindelöf [Mat, p.88].  DCCC implies Property ($\varepsilon$) even without $T_3$ assumption. Also it seems that the implication (“countably compact” $\to$ “almost $\omega_1$-Lindelöf”) at the first diagram is wrong, and the cover $\{[0,\alpha): \alpha<\omega_1\}$ of an ordinal $\omega_1$ endowed with the order topology is a counterexample (as Matveev wrote too at the beginning of the page 93). So Property A is incomparable with pseudocompactness, because also each Lindelöf space has Property A. A space $X$ is linearly Lindelöf if every open cover of $X$, linearly ordered by the subset relation, has a countable subcover. Clearly, each linearly Lindelöf space has Property A too.
References 
[BR, Sect. 1.2] Taras Banakh, Alex Ravsky. Verbal covering properties of topological spaces // arXiv: 1503.04480 [math.GN], Topology Appl. (Proceedings of Lepanto Conference) (to be published).
[DRRT, Sect. 3.1]  E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree, Star covering properties. Topology Appl., 39:1 (1991), 71-103.
[Mat, Ch. 5] M. Matveev, A Survey on Star Covering Properties.
( The propoerty (a) considered in Matveev’s paper differs from yours peoperty A. :-) )
[Par] C. M. Pareek, On some generalizations of countably compact and Lindelöf spaces, Suppl. Rend. Circ. Mat. di Palermo, Ser II, 24 (1990) 169-192.
