Manifold pathologies and Baire-1 functions as cohomotopy groups

A slight extension of a construction due to Calabi-Rosenlicht (and later Gabard, Baillif and others) produces a typically non-metrizable $n$-manifold $\mathbb{P}$ by gluing two copies of the open upper half-space ${\mathbb{H}}_{++}$ in ${\mathbb{R}}^n$ along the disjoint union of the spaces of rays within ${\mathbb{H}}_{++}$ originating at points ranging over a subset $S\subseteq {\mathbb{R}}^{n-1}$ of the boundary ${\mathbb{R}}^{n-1}=\mathrm{\partial }\overline{{\mathbb{H}}_{++}}$. The fundamental group ${\pi }_1(\mathbb{P})$ is free on the complement $S^{\times }$ of any singleton in $S\mathrm{\ne }\mathrm{\varnothing }$, and the main result below is that the first cohomotopy group ${\pi }^1(\mathbb{P})$, regarded as a space of functions $S^{\times }\to \mathbb{Z}$, is precisely the additive group of integer-valued Baire-1 functions on $S^{\times }$. This occasions a detour on characterizations (perhaps of independent interest) of Baire-1 real-valued functions on a metric space $(B,d)$ as various types of non-tangential boundary limits of continuous functions on $B\times {\mathbb{R}}_{>0}$.