# When does the refinement of a paracompact topology remain paracompact?

Let $$(X,\tau)$$ be a Hausdorff paracompact space. Let $$\tau'$$ be the smallest $$P$$-topology refining $$(X,\tau)$$, i.e. the topology which has for base the $$G_\delta$$-subsets of $$(X,\tau)$$.

Is it true that $$(X,\tau')$$ is again paracompact?

If not, does the result hold under extra assumptions (e.g. hereditary paracompactness or even more)?

Is there a Hausdorff paracompact space $$(X,\tau)$$ where every point has a local base of size $$\leq \omega_1$$ such that $$(X,\tau')$$ is not paracompact?

• The space $(X,\tau')$ is commonly known as the $P$-space coreflection of $(X,\tau)$. Commented Oct 2, 2022 at 15:51

If $$\kappa$$ is an infinite cardinal, then the $$<\kappa$$-box topology on a product $$\prod_{i\in I}X_i$$ of topological spaces is the topology where the basic open sets are of the form $$\prod_{i\in I}U_i$$ where each $$U_i$$ is an open subset of $$X_i$$ and where $$|\{i\in I\mid U_i\neq X_i\}|<\kappa$$. The $$P$$-space coreflection of a space $$(X,\tau)$$ is the space $$(X,\tau')$$ where $$\tau'$$ is generated by the $$G_\delta$$ subsets of $$(X,\tau)$$.
Theorem: If $$\kappa$$ is an infinite cardinal, $$\kappa$$ is given the discrete topology, and $$\kappa^{(\kappa^+)}$$ is given the $$<\kappa$$-box topology, then $$\kappa^{(\kappa^+)}$$ is not normal.
In particular, if we assume the continuum hypothesis, then $$\mathfrak{c}^{(\mathfrak{c}^+)}$$ is not normal, but $$\mathfrak{c}^{(\mathfrak{c}^+)}$$ is the $$P$$-space coreflection of the compact space $$[0,1]^{\mathfrak{c}^+}$$ with the product topology.
• Thank you! That's more than I was hoping for. By the way, do you know by chance whether there is also a counterexample where every point has a local base of size $\leq\omega_1$?
• I am not sure about how to get a space with a local basis of size at most $\aleph_1$. Commented Oct 5, 2022 at 12:55