Some local network properties connected with convergence of sequences are preserved by countable box-products. For example, if topological spaces $X_n$, $n\in\omega$, have countable $cs$-networks at points $x_n\in X_n$, then the box-product $\square_{n\in\omega}X_n$ has a countable $cs$-network at the point $(x_n)_{n\in\omega}$.

We recall that a family $\mathcal N$ of subsets of a topological space $X$ is a $cs$-network at a point $x\in X$ 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 $N\in\mathcal N$ such that $N\subset O_x$ and $N$ contains all but finitely many elements of the sequence $(x_n)$.

Another property preserved by countable box-product is the existence of an $\omega^\omega$-base at each point.

We recall that a topological space $X$ has an $\omega^\omega$-base at a point $x\in X$ if there exists a neighborhood base $(U_\alpha)_{\alpha\in\omega^\omega}$ at $x$ indexed by elements of $\omega^\omega$ so that $U_\alpha\subset U_\beta$ for any $\beta\le\alpha$ in $\omega^\omega$.

Spaces with $\omega^\omega$-base are extensively studied in this paper and often appear in Topological Algebra and Functional Analysis.

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 $x$).