Although the box topology is a topology worth studying and is similar to the strong topology in differential topology, the box topology is in many regards very badly behaved since the box product of even nice spaces has many undesirable properties. For instance, unlike ordinary products, non-trivial box products are never connected, metrizable, nor locally compact. I want to know what properties of topological spaces are preserved under taking box products. Let me list the more obvious properties.


$\textbf{Preserved under box products:}$
Separation axioms such as $T\_{0},T\_{1}$, Hausdorffness, regularity, complete regularity; zero-dimensionality, total disconnectedness(otherwise known as hereditary disconnectedness), total separation (two distinct points can always be separated by a clopen set), other disconnectedness properties, negative properties in general, $P$-spaces, discreteness, complete uniformizability.


$\textbf{Not preserved under box products:}$ Normality, paracompactness, ultraparacompactness, compactness, other compactness properties, connectedness, path connectedness, other connectedness properties, . . .


See chapter 4 in the Handbook of set-theoretic topology for more information on box products and a few other properties preserved under box products that I neglected to mention in this question.


For which other topological properties $P$ does it hold that if $X\_{i}$ has property $P$ for all $i\in I$, then $\prod\_{i\in I}^{\textrm{Box}}X\_{i}$ also has property $P$? I am looking for specific examples of such properties since I do not believe that there is a nice general characterization of all such properties. I am looking for positive properties rather than the negation of certain well known properties.



Which topological properties are preserved under taking box products?
A space $X$ is discretely generated (DG) if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is a discrete set $D \subset A$ such that $x \in \overline{D}$. 


For being a convergence-type property (note that Fréchet-Urysohn and even radial spaces are DG) discrete generability is pretty weak: every scattered space is DG, every compact space of countable tightness is DG in ZFC and every monotonically normal space is DG (see *Dow, A.; Tkachenko, M.G.; Tkachuk, V.V.; Wilson, R.G.*, Topologies generated by discrete subspaces, Glas. Mat., III. Ser. 37, No.1, 187-210 (2002). [ZBL1009.54005](https://zbmath.org/?q=an:1009.54005).).


There are examples of DG spaces with a non-DG square (see *Murtinová, Eva*, [**On products of discretely generated spaces**](http://dx.doi.org/10.1016/j.topol.2005.11.014), Topology Appl. 153, No. 18, 3402-3408 (2006). [ZBL1107.54006](https://zbmath.org/?q=an:1107.54006).), but in a few notable special case discrete generability is indeed preserved in box products:



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> > THEOREM (Tkachuk, 2012): Every box product of monotonically normal spaces is discretely generated.
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> > THEOREM (Barriga-Acosta and Hernández-Hernández, 2016): Every box product of regular first-countable spaces is discretely generated.
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