Which topological properties are preserved under taking box products?

Question

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.

Answer 1

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.).

There are examples of DG spaces with a non-DG square (see Murtinová, Eva, On products of discretely generated spaces, Topology Appl. 153, No. 18, 3402-3408 (2006). ZBL1107.54006.), but in a few notable special case discrete generability is indeed preserved in box products:

THEOREM (Tkachuk, 2012): Every box product of monotonically normal spaces is discretely generated.

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

Answer 2

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 $x$).

Answer 3

A space is said to be hyperconnected if every two non-empty open sets have non-empty intersection.

The axiom of choice implies that the box product of hyperconnected spaces is hyperconnected.

Answer 4

A space is called Alexandrov if every open set is clopen. If $X_i$ are Alexandrov spaces, then so is $\prod_{i\in I}^{\textrm{Box}}X_{i}$.

Answer 5

Caution: Answer contains a serious mistake (see comments below by Joseph van Name), but is not deleted for the record.

A space $(X,\tau)$ 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 $X$. This is a survey of $D$-spaces and their properties.

A routine verification shows that whenever $X_i$ are $D$-spaces for all $i$, then so is $\prod_{i\in I}^{\textrm{Box}}X_{i}$.

EDIT: Verification of the claim above.

Let $X_i, i\in I$ be $D$-spaces and assume $\{N\big((x_i)\big):(x_i) \in\prod_{i\in I} X_i\}$ is an assignment of a neighborhood to each point of $\prod_{i\in I} X_i$ with the box product topology. We can assume that $N\big((x_i)\big) = \prod_{i\in I} N_i(x_i)$ for each $i$ where $N_i(x_i)$ is an open neighborhood of $x_i\in X_i$. Since each $X_i$ is a $D$-space let $D_i$ be a discrete set in $X_i$ such that $\bigcup \{N_i(d): d\in D_i\} = X_i$, for each $i$. So we set $$D = \prod_{i\in I} D_i.$$

Let $(y_i)\in \prod_{i\in I} X_i$. For each $i\in I$ we have $y_i \in N_i(d_i)$ for some $d_i\in X_i$ because by assumption $\bigcup \{N_i(d): d\in D_i\} = X_i$. So let $d:= (d_i)$, so $d\in D$. Then $(y_i) \in N\big((d_i)\big)$. This implies that $$\bigcup \{N\big((x_i)\big): (x_i)\in D\} = \prod_{i\in I} X_i.$$ Therefore $\prod_{i\in I} X_i$ with the box product topology is a $D$-space.