considered as a subspace of the compact-open topology on the Cartesian power $X^{\mathbb{R}}$.

A locally compact space $X$ satisfies the MOP if and only if $C_k(X)$ is Baire, i.e. satisfies the Baire Category Theorem.

\[lem6\] Locally compact, paracompact spaces satisfy the MOP.

\[thm35\] ${\mathrm}{MM}(S)[S]$ implies that normal spaces satisfying the MOP are paracompact if they are:

-   locally compact, countably tight, and hereditarily normal, or

-   first countable and hereditarily normal, or

-   locally compact, countably tight with Lindelöf number $\leq\aleph_1$, or

-   first countable, with Lindelöf number $\leq\aleph_1$, or

-   locally compact, countably tight, and countable sets have Lindelöf closures.

These all follow easily from \[thmConj2\], \[thm312\], and **Moore-Mrówka**, using:

In a sequential space, countably compact subspaces are closed.

Countably compact spaces satisfying the MOP are compact.

\[lem510\] First countable spaces satisfying the MOP are locally compact.

The one-point compactification of a locally compact space $X$ is countably tight if and only if $X$ does not include a perfect pre-image of $\omega_1$.

If they have the MOP, sequential spaces do not include copies of $\omega_1$, so (1) follows from \[312\]. (2) follows from (1) plus \[lem510\]. (3) follows from \[lem512\] plus \[thmConj2\]. (4) follows from (3) plus \[lem510\]. (5) follows from \[thm312\], \[thmConj2\] and Balogh’s Lemma above.

In the special case of a space with the MOP, we have:

\[thm513\] If ZFC is consistent, then it is consistent to additionally assume that first countable normal spaces satisfying the MOP and with Lindelöf number $\leq\aleph_1$ are paracompact.

Such a space is locally compact and does not include a perfect pre-image of $\omega_1$.

MA$_{\omega_1}$ gives counterexamples for the conclusions of \[thm35\] and \[thm513\]. See e.g. [@T5].

If ZFC is consistent, then it is consistent to assume that first countable hereditarily normal, locally connected spaces satisfying the MOP a