Is every $k_\omega$ space a $k$ space?

Question

A $k_\omega$ space is the union of compacts $K_n$ such that a set is closed if and only if its intersection with each $K_n$ is closed within that subspace.

A $k$ space has the property that a set is closed if and only if its intersection with each compact is closed within that subspace.

Is it true that every $k_\omega$ space is also a $k$ space?

Answer 1

Tyrone's answer provides more information, but it seems the following direct argument works.

Suppose $X$ is a $k_\omega$ space with respect to the $K_n$. To show that $X$ is a k-space, take $A\subseteq X$ such that $A\cap K$ is closed in $K$ for each compact $K\subseteq X$. Then in particular each $A\cap K_n$ is closed in $K_n$. By the $k_\omega$ property, $A$ is closed in $X$.

Answer 2

Yes. Every $k_\omega$-space is the quotient of a countable sum of compact spaces. This sum is a $k$-space, as is any quotient of it.

In more detail we consider the following situation.

Let $X$ be a space and suppose that $\mathcal{G}$ is a covering of $X$ (by arbitrary subspaces). The following statements are equivalent.

1. The topology of $X$ is determined by $\mathcal{G}$ in the sense that a subspace $A\subseteq X$ is closed (open) if and only if $A\cap G$ is relatively closed (open) in $G$ for each $G\in\mathcal{G}$.
2. The canonical map $\bigsqcup_{G\in\mathcal{G}}G\rightarrow X$ is a quotient mapping, where the domain is the topological sum of the members of $\mathcal{G}$, and the mapping takes each summand directly onto its image in $X$.

The proof of this observation is clear.

In any case, suppose that $\mathscr{P}$ is a topological property such that $(i)$ the one-point space has $\mathscr{P}$, and $(ii)$ if $X$ is any space with $\mathscr{P}$, then every quotient of $X$ has $\mathscr{P}$. We consider the category $\mathcal{C}_\mathscr{P}$ consisting of all spaces whose topologies are determined by the covering consisting of all their subspaces with $\mathscr{P}$.

The equivalent properties of coverings described above makes the following clear.

• $\mathcal{C}_\mathscr{P}$ is closed under topological sums.
• $\mathcal{C}_\mathscr{P}$ is closed under quotient images.
• Every space with $\mathscr{P}$ belongs to $\mathcal{C}_\mathscr{P}$.
• If $X$ is determined by a covering $\mathcal{G}$ consisting of spaces in $\mathcal{C}_\mathscr{P}$, then $X$ is is $\mathcal{C}_\mathscr{P}$.

We apply this with $\mathscr{P}$ being the property 'compact', in which case $\mathcal{C}_\mathscr{P}$ is the category of $k$-spaces. By the above, any sum of k-spaces is a $k$-space, and any quotient of a $k$-space is another k-space.

The point of the the last bullets is that it is not necessary to describe a k-space by means of the covering consisting of all its compact subspaces. Any determining covering consisting of compact subspaces is sufficient. As such, a $k_\omega$-space is a space with a countable determining covering consisting of compact subspaces. Any such space is a k-space.

We could also consider other examples. For instance, $(i)$ $k_\alpha$-spaces for a cardinal $\alpha$ as those spaces determined by a covering of cardinality $\leq\alpha$ consisting of compact subspaces $(ii)$ $s_\omega$-spaces as those spaces determined by a countable covering of compact metric subspaces. Any $k_\alpha$ is a $k$-space and any $s_\omega$-space is sequential.