TITLE: Which topological properties are preserved under taking box products?
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: Some local network or base properties are preserved by countable box-products of topological spaces. 
In particular:
1) the existence of a countable $cs$-network at each point;
2) the existence of a countable $cs^*$-network at each point;
3) the existence of a countable $s^*$-network at each point;
4) the existence of an $\omega^\omega$-base at each point.
Now I recall the corresponding definitions.
Let $X$ be a topological space and $x$ be a point of $X$. A family $\mathcal F$ of subsets of $X$ is called a 
$\bullet$ a $cs$-network at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ convergent to $x$  and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains all but finitely many elements of the sequence $(x_n)$;
$\bullet$ a $cs^*$-network at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ convergent to $x$  and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains infinitely many elements of the sequence $(x_n)$;
$\bullet$ an $s^*$-network at $x$ if for any sequence $\{x_n\}_{n\in\omega}\subset X$ accumulating at $x$  and any neighborhood $O_x\subset X$ of $x$ there exists a set $F\in\mathcal F$ such that $F\subset O_x$ and $F$ contains infinitely many elements of the sequence $(x_n)$;
$\bullet$ an $\omega^\omega$-base at $x$ if each set $F\in\mathcal F$ is a neighborhood of $x$ and $\mathcal F$ can be written as $\mathcal F=\{F_\alpha\}_{\alpha\in\omega^\omega}$ so that $F_\alpha\subset F_\beta$ for any elements $\beta\le\alpha$ of $\omega^\omega$ (endowed with the ccordinatewise partial order).
Those notions are studied in this paper and often appear in Topological Algebra and Functional Analysis. 
For a topological space $X$ and a point $x\in X$ we have the implications:
($X$ has an $\omega^\omega$-base at $x$) $\Rightarrow$ ($X$ has a countable $s^*$-network at $x$) $\Rightarrow$ 
$\Rightarrow$ ($X$ has a countable $cs^*$-network at $x$) $\Leftrightarrow$ ($X$ has a countable $cs$-network at $x$).