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For a cardinal $\kappa$ let $\Box^\kappa\mathbb R$ be the box-product of $\kappa$-many lines and $\boxdot^\kappa\mathbb R:=\{x\in\Box^\kappa:|\{\alpha\in\kappa:x(\alpha)\ne 0\}<\omega\}$ be the $\sigma$-product of $\kappa$ many lines in $\Box^\kappa\mathbb R$.

The box-product $\Box^\kappa\mathbb R$ carries a natural structure of an Abelian topological group and $\boxdot^\kappa\mathbb R$ is a subgroup of $\Box^\kappa\mathbb R$.

Let us call a topological space $X$ box-flat at a point $x\in X$ if for each cardinal $\kappa$ and each continuous map $f:X\to \Box^\kappa\mathbb R$ the point $x$ has a neighborhood $U_x\subset X$ whose image $f(U_x)$ is contained in the coset $f(x)+\boxdot^\kappa\mathbb R$.

A topological space $X$ is called box-flat if it is box-flat at each point $x\in X$.

It can be shown that a topological space $X$ is box-flat at a point $x\in X$ if one of the following three conditions is satisfied:

  • $X$ is first-countable at $x$;
  • $x$ has a connected neighborhood in $X$;
  • $x$ has a countably compact neighborhood in $X$.

What these three properties have in common (besides the box-flatteness)?

Question. Is there any known class of topological spaces that contains all first-countable spaces and all (locally) connected spaces, all (locally) compact spaces and is contained in the class of box-flat spaces?

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