# Manifold pathologies and Baire-1 functions as cohomotopy groups

Alexandru Chirvasitu

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Alexandru Chirvasitu. Manifold pathologies and Baire-1 functions as cohomotopy groups. 2024. hal-04571518

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# Manifold pathologies and Baire-1 functions as cohomotopy groups

Alexandru Chirvasitu

► To cite this version:

Alexandru Chirvasitu. Manifold pathologies and Baire-1 functions as cohomotopy groups. 2024. hal-04571518

HAL Id: hal-04571518

https://hal.science/hal-04571518v1

Preprint submitted on 8 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# Manifold pathologies and Baire-1 functions as cohomotopy groups

Alexandru Chirvasitu

## Abstract

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 $\mathbb{R}^{n-1} = \partial \overline{\mathbb{H}_{++}}$. The fundamental group $\pi_1(\mathbb{P})$ is free on the complement $S^\times$ of any singleton in $S \neq \emptyset$, 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 $B \times \mathbb{R}_{>0}$.

*Key words: Baire one; Bruschlinsky group; Lipschitz; Prüfer manifold; Prüfer surface; boundary; cohomotopy group; dilatation; fundamental group; gauge; germ; non-tangential limit;*

MSC 2020: 26A21; 55Q55; 57N65; 54E52; 54C35; 26A16; 55Q05; 55Q52

## Contents

1 1-cohomotopy for boundary-less Prüfer manifolds 3

2 Non-tangential limits and Borel-1 functions 7

References 12

## Introduction

The *Prüfer surface* ($\mathbb{P}$ throughout the rest of the present section) of [15, §3] appears to have been initially introduced in [11, §3] (where it is denoted by $S$) as auxiliary in those authors' verification of a conjecture of Bochner's to the effect that there are non-second-countable (so non-metrizable, or equivalently [30, Corollary 2.4], non-*paracompact* [39, Definition 20.6]) complex-analytic manifolds of complex dimension $> 1$. Prüfer surfaces and their cousins exemplify various aspects of the rich subject of non-metrizable manifolds: see e.g. the monograph [16], [2, 3, 4, 6, 13, 14, 30, 31, 34] and their many references, etc.

The description of $\mathbb{P}$ given in [15, §3] is what Definition 1.1 below builds on:

* Consider the open upper half-plane $\mathbb{H}_{++} \subset \mathbb{R}^2$, with its usual topology;

* Supplement $\mathbb{H}_{++}$ with a $\mathbb{R}$-parametrized family of boundary lines $\mathbb{R}$, with the boundary component $\mathbb{R}_p \cong \mathbb{R}$ with parameter

$$p \in \mathbb{R} := \text{the usual } x\text{-axis } \{(x, 0) \mid x \in \mathbb{R}\} \subset \mathbb{R}^2$$
# Manifold pathologies and Baire-1 functions as cohomotopy groups

Alexandru Chirvasitu

## Abstract

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 $\mathbb{R}^{n-1} = \partial \overline{\mathbb{H}_{++}}$. The fundamental group $\pi_1(\mathbb{P})$ is free on the complement $S^\times$ of any singleton in $S \neq \emptyset$, 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 $B \times \mathbb{R}_{>0}$.

*Key words: Baire one; Bruschlinsky group; Lipschitz; Prüfer manifold; Prüfer surface; boundary; cohomotopy group; dilatation; fundamental group; gauge; germ; non-tangential limit;*

MSC 2020: 26A21; 55Q55; 57N65; 54E52; 54C35; 26A16; 55Q05; 55Q52

## Contents

1 1-cohomotopy for boundary-less Prüfer manifolds 3

2 Non-tangential limits and Borel-1 functions 7

References 12

## Introduction

The *Prüfer surface* ($\mathbb{P}$ throughout the rest of the present section) of [15, §3] appears to have been initially introduced in [11, §3] (where it is denoted by $S$) as auxiliary in those authors' verification of a conjecture of Bochner's to the effect that there are non-second-countable (so non-metrizable, or equivalently [30, Corollary 2.4], non-*paracompact* [39, Definition 20.6]) complex-analytic manifolds of complex dimension $> 1$. Prüfer surfaces and their cousins exemplify various aspects of the rich subject of non-metrizable manifolds: see e.g. the monograph [16], [2, 3, 4, 6, 13, 14, 30, 31, 34] and their many references, etc.

The description of $\mathbb{P}$ given in [15, §3] is what Definition 1.1 below builds on:

* Consider the open upper half-plane $\mathbb{H}_{++} \subset \mathbb{R}^2$, with its usual topology;

* Supplement $\mathbb{H}_{++}$ with a $\mathbb{R}$-parametrized family of boundary lines $\mathbb{R}$, with the boundary component $\mathbb{R}_p \cong \mathbb{R}$ with parameter

$$p \in \mathbb{R} := \text{the usual } x\text{-axis } \{(x, 0) \mid x \in \mathbb{R}\} \subset \mathbb{R}^2$$
consisting of the rays in the upper half-plane based at $p$.

• The topology on $\mathbb{R}_p$ is the guessable ("angular") one, with ray nearness measured by angle inclination, whereas a net $(x_\lambda) \subset \mathbb{H}_{++}$ converges to a ray $r \in \mathbb{R}_p$ if it converges to the point $p \in \mathbb{R} \subset \mathbb{R}^2$ in the usual sense, and in addition the rays $\overrightarrow{px_\lambda}$ converge to $r$ in the aforementioned angular topology of $\mathbb{R}_p$.

This thus far will give a 2-manifold $\mathbb{P}_\partial$ with boundary

$$ \partial \mathbb{P}_\partial = \coprod_{p \in \mathbb{R}} \mathbb{R}_p; \qquad (0-1) $$

here and throughout the paper an $n$-manifold is a Hausdorff space locally homeomorphic to either $\mathbb{R}^n$ or the closed upper half-space therein (as in [3, Introduction], for example).

• Finally, $\mathbb{P}$ is the double [27, Example 9.32]

$$ \mathbb{P} := 2\mathbb{P}_\partial := \mathbb{P}_\partial \coprod_{\partial\mathbb{P}_\partial} \mathbb{P}_\partial. $$

This produces a connected (real-analytic, even [11, §3]) boundary-less 2-manifold, whose fundamental group is [15, Proposition 3] free on continuum many generators, naturally parametrized in the course of that proof by the subset $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$ of the space $\mathbb{R}$ housing the parameters $p$ of (0-1).

On the same homotopy-theoretic theme, recall the first *cohomotopy* [21, §VII.1] or *Bruschlinsky group* ([21, §II.7], [25, pre §1])

$$ \pi^1(X) := [X, \mathbb{S}^1] := \text{free homotopy classes of continuous maps } X \to \mathbb{S}^1 \qquad (0-2) $$

of a space $X$. There is generally an obvious pairing (or *comparison*) morphism [21, §VII.1]

$$ \pi^1(X) \xrightarrow{\text{COMP}} \text{Hom}(\pi_1(X), \mathbb{Z} \cong \pi_1(\mathbb{S}^1)), \quad (X \text{ path-connected}) \qquad (0-3) $$

obtained by composing maps $\mathbb{S}^1 \to X$ and $X \to \mathbb{S}^1$ into loops in $\mathbb{S}^1$, and for locally path-connected $X$ (such as the manifold $\mathbb{P}$) $\text{COMP}$ is injective: continuous maps $X \to \mathbb{S}^1$ trivial on $\pi_1$ lift [29, Lemma 79.1] through the universal cover $\mathbb{R} \to \mathbb{S}^1$ and hence homotope to constants. Recalling [24, Definition 24.1] that the *Baire-1 functions* between metrizable spaces are those which pull back open sets to countable unions of closed sets, one small sample of the main result in Theorem 1.5 below is as follows:

**Theorem A** *The Bruschlinsky subgroup*

$$ \pi^1(\mathbb{P}) \xrightarrow{\text{COMP}} \text{Hom}(\pi_1(\mathbb{P}), \mathbb{Z}) \cong \mathbb{Z}^{\mathbb{R}^\times} $$

of the *Prüfer surface* is precisely the *additive group* of *Baire-1 functions* $\mathbb{R}^\times \to \mathbb{Z}$.

In the process of unwinding one of the implications (realizing every Baire-1 function as a co-homotopy class) a few alternative characterizations of the Baire-1 property emerge. Once more compressing and trimming for the purpose of illustration (Theorems 2.2 and 2.5):

**Theorem B** *For a function $B \xrightarrow{f} \mathbb{R}$ on a metric space $(B, d)$ the following conditions are equivalent.*
consisting of the rays in the upper half-plane based at $p$.

• The topology on $\mathbb{R}_p$ is the guessable ("angular") one, with ray nearness measured by angle inclination, whereas a net $(x_\lambda) \subset \mathbb{H}_{++}$ converges to a ray $r \in \mathbb{R}_p$ if it converges to the point $p \in \mathbb{R} \subset \mathbb{R}^2$ in the usual sense, and in addition the rays $\overrightarrow{px_\lambda}$ converge to $r$ in the aforementioned angular topology of $\mathbb{R}_p$.

This thus far will give a 2-manifold $\mathbb{P}_\partial$ with boundary

$$ \partial \mathbb{P}_\partial = \coprod_{p \in \mathbb{R}} \mathbb{R}_p; \qquad (0-1) $$

here and throughout the paper an $n$-manifold is a Hausdorff space locally homeomorphic to either $\mathbb{R}^n$ or the closed upper half-space therein (as in [3, Introduction], for example).

• Finally, $\mathbb{P}$ is the double [27, Example 9.32]

$$ \mathbb{P} := 2\mathbb{P}_\partial := \mathbb{P}_\partial \coprod_{\partial\mathbb{P}_\partial} \mathbb{P}_\partial. $$

This produces a connected (real-analytic, even [11, §3]) boundary-less 2-manifold, whose fundamental group is [15, Proposition 3] free on continuum many generators, naturally parametrized in the course of that proof by the subset $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$ of the space $\mathbb{R}$ housing the parameters $p$ of (0-1).

On the same homotopy-theoretic theme, recall the first *cohomotopy* [21, §VII.1] or *Bruschlinsky group* ([21, §II.7], [25, pre §1])

$$ \pi^1(X) := [X, \mathbb{S}^1] := \text{free homotopy classes of continuous maps } X \to \mathbb{S}^1 \qquad (0-2) $$

of a space $X$. There is generally an obvious pairing (or *comparison*) morphism [21, §VII.1]

$$ \pi^1(X) \xrightarrow{\text{COMP}} \text{Hom}(\pi_1(X), \mathbb{Z} \cong \pi_1(\mathbb{S}^1)), \quad (X \text{ path-connected}) \qquad (0-3) $$

obtained by composing maps $\mathbb{S}^1 \to X$ and $X \to \mathbb{S}^1$ into loops in $\mathbb{S}^1$, and for locally path-connected $X$ (such as the manifold $\mathbb{P}$) $\text{COMP}$ is injective: continuous maps $X \to \mathbb{S}^1$ trivial on $\pi_1$ lift [29, Lemma 79.1] through the universal cover $\mathbb{R} \to \mathbb{S}^1$ and hence homotope to constants. Recalling [24, Definition 24.1] that the *Baire-1 functions* between metrizable spaces are those which pull back open sets to countable unions of closed sets, one small sample of the main result in Theorem 1.5 below is as follows:

**Theorem A** *The Bruschlinsky subgroup*

$$ \pi^1(\mathbb{P}) \xrightarrow{\text{COMP}} \text{Hom}(\pi_1(\mathbb{P}), \mathbb{Z}) \cong \mathbb{Z}^{\mathbb{R}^\times} $$

of the *Prüfer surface* is precisely the *additive group* of *Baire-1 functions* $\mathbb{R}^\times \to \mathbb{Z}$.

In the process of unwinding one of the implications (realizing every Baire-1 function as a co-homotopy class) a few alternative characterizations of the Baire-1 property emerge. Once more compressing and trimming for the purpose of illustration (Theorems 2.2 and 2.5):

**Theorem B** *For a function $B \xrightarrow{f} \mathbb{R}$ on a metric space $(B, d)$ the following conditions are equivalent.*
(a) $f$ is Baire-1.

(b) $f$ is the non-tangential limit along $B \cong B \times \{0\}$ of a continuous function $B \times \mathbb{R}_{>0} \xrightarrow{\overline{f}} \mathbb{R}$ in the sense that

$$ \overline{f}(b', t) \xrightarrow[b' \to b, t \to 0]^{d(b,b') \le Ct} f(b), \quad \forall b \in B, \forall C > 0. \quad (0-4) $$

(c) $f$ is the non-tangential limit along $B \cong B \times \{0\} \subset B' \times \{0\}$ of a continuous function $B' \times \mathbb{R}_{>0} \xrightarrow{\overline{f}} \mathbb{R}$ in the sense that (0-4) holds with $b'$ ranging over some (equivalently, any) metric space $B'$ containing $(B, d)$ isometrically. ■

## Acknowledgements

This work is partially supported by NSF grant DMS-2001128.

# 1 1-cohomotopy for boundary-less Prüfer manifolds

In their two-dimensional incarnations, the following constructions recover objects variously referred to as *Prüfer surfaces* (so named because [33] credits Prüfer for the original idea). They are very much in the spirit of other generalizations in the literature, e.g. the *Prüferization* procedure of [14, Appendix B].

**Definition 1.1** Having fixed a positive integer $n \ge 2$ throughout, write

$$ \mathbb{H}_{++} = \mathbb{H}_{++}^n := \text{strict upper half-space } \{(x_1 \cdots x_n) \in \mathbb{R}^n \mid x_n > 0\} \text{ and} \\ \mathbb{H}_+ = \mathbb{H}_+^n := \text{closure } \overline{\mathbb{H}_{++}} \text{ in } \mathbb{R}^n = \{(x_1 \cdots x_n) \in \mathbb{R}^2 \mid x_n \ge 0\}. \quad (1-1) $$

(1) For a subset $S \subseteq \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$ the *bordered Prüfer n-manifold* $\mathbb{P}\mathbb{F}_{S,\partial} = \mathbb{P}\mathbb{F}_{S,\partial}^n$ is obtained by

* setting

$$ \mathbb{P}\mathbb{F}_{S,\partial} := \mathbb{H}_{++} \cup \coprod_{p \in F} \mathcal{R}_p, \quad \mathcal{R}_p := \{\text{rays in } \mathbb{H}_{++} \text{ originating at } p\} $$

as a set (the rays have $p$ as an origin, apart from which they lie entirely *above* the hyperplane $\mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$);

* topologizing $\mathbb{H}_{++}$ as usual, with its subspace topology inherited from $\mathbb{R}^n$;

* topologizing each $\mathcal{R}_p$ as a subspace of the projective space $\mathbb{P}^{n-1}$ of lines in $\mathbb{R}^n$ passing through $p$;

* and having a net $(x_\lambda)_\lambda \subset \mathbb{H}_{++}$ converge to $r \in \mathcal{R}_p \subset \mathbb{P}\mathbb{F}_{S,\partial}$ (language: $(x_\lambda)$ converges tangentially to $r$) precisely when

$$ \begin{gathered} x_\lambda \xrightarrow{\lambda} p \text{ in the usual plane topology and also} \\ \overline{px_\lambda} \xrightarrow{\lambda} r \text{ in } \mathcal{R}_p. \end{gathered} $$

$\mathbb{P}\mathbb{F}_{\partial}^2 = \mathbb{P}\mathbb{F}_{\mathbb{R},\partial}^2$ is the surface $P_0$ of [15, §3], $P_s$ of [2, §2] and what [14, Appendix B] would denote by $P_H(H)$. More generally, for arbitrary $Y \subseteq \mathbb{R}$, $\mathbb{P}\mathbb{F}_{Y,\partial}^2$ is the $P_Y(H)$ of loc. cit. We drop the ‘F’ subscript when it happens to be the entire boundary $\mathbb{R}^{n-1}$ of $\mathbb{H}_+$.
(a) $f$ is Baire-1.

(b) $f$ is the non-tangential limit along $B \cong B \times \{0\}$ of a continuous function $B \times \mathbb{R}_{>0} \xrightarrow{\overline{f}} \mathbb{R}$ in the sense that

$$ \overline{f}(b', t) \xrightarrow[b' \to b, t \to 0]^{d(b,b') \le Ct} f(b), \quad \forall b \in B, \forall C > 0. \qquad (0-4) $$

(c) $f$ is the non-tangential limit along $B \cong B \times \{0\} \subset B' \times \{0\}$ of a continuous function $B' \times \mathbb{R}_{>0} \xrightarrow{\overline{f}} \mathbb{R}$ in the sense that (0-4) holds with $b'$ ranging over some (equivalently, any) metric space $B'$ containing $(B, d)$ isometrically. ■

## Acknowledgements

This work is partially supported by NSF grant DMS-2001128.

# 1 1-cohomotopy for boundary-less Prüfer manifolds

In their two-dimensional incarnations, the following constructions recover objects variously referred to as *Prüfer surfaces* (so named because [33] credits Prüfer for the original idea). They are very much in the spirit of other generalizations in the literature, e.g. the *Prüferization* procedure of [14, Appendix B].

**Definition 1.1** Having fixed a positive integer $n \ge 2$ throughout, write

$$ \mathbb{H}_{++} = \mathbb{H}_{++}^n := \text{strict upper half-space } \{(x_1 \cdots x_n) \in \mathbb{R}^n \mid x_n > 0\} \text{ and} \\ \mathbb{H}_+ = \mathbb{H}_+^n := \text{closure } \overline{\mathbb{H}_{++}} \text{ in } \mathbb{R}^n = \{(x_1 \cdots x_n) \in \mathbb{R}^2 \mid x_n \ge 0\}. \qquad (1-1) $$

(1) For a subset $S \subseteq \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$ the *bordered Prüfer n-manifold* $\mathbb{P}\mathbb{F}_{S,\partial} = \mathbb{P}\mathbb{F}_{S,\partial}^n$ is obtained by

* setting

$$ \mathbb{P}\mathbb{F}_{S,\partial} := \mathbb{H}_{++} \sqcup \coprod_{p \in F} \mathcal{R}_p, \quad \mathcal{R}_p := \{\text{rays in } \mathbb{H}_{++} \text{ originating at } p\} $$

as a set (the rays have $p$ as an origin, apart from which they lie entirely above the hyperplane $\mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$);

* topologizing $\mathbb{H}_{++}$ as usual, with its subspace topology inherited from $\mathbb{R}^n$;

* topologizing each $\mathcal{R}_p$ as a subspace of the projective space $\mathbb{P}^{n-1}$ of lines in $\mathbb{R}^n$ passing through $p$;

* and having a net $(x_\lambda)_\lambda \subset \mathbb{H}_{++}$ converge to $r \in \mathcal{R}_p \subset \mathbb{P}\mathbb{F}_{S,\partial}$ (language: $(x_\lambda)$ converges tangentially to $r$) precisely when

$$ \begin{gathered} x_\lambda \xrightarrow{\lambda} p \text{ in the usual plane topology and also} \\ \overline{px_\lambda} \xrightarrow{\lambda} r \text{ in } \mathcal{R}_p. \end{gathered} $$

$\mathbb{P}\mathbb{F}_{\partial}^2 = \mathbb{P}\mathbb{F}_{\mathbb{R},\partial}^2$ is the surface $P_0$ of [15, §3], $P_s$ of [2, §2] and what [14, Appendix B] would denote by $P_H(H)$. More generally, for arbitrary $Y \subseteq \mathbb{R}$, $\mathbb{P}\mathbb{F}_{Y,\partial}^2$ is the $P_Y(H)$ of loc. cit. We drop the ‘F’ subscript when it happens to be the entire boundary $\mathbb{R}^{n-1}$ of $\mathbb{H}_+$.
(2) The (plain, or double, or boundary-less) Prüfer n-manifold PF<sub>S</sub> = PF<sub>S</sub><sup>n</sup> is the double 2PF<sub>S,∂</sub> obtained as usual (e.g. [26, Example 3.80]), by gluing two copies of PF<sub>S,∂</sub> along its boundary:

$$ \mathrm{PF}_S^n = \mathrm{PF}_{S,\partial} \coprod_{\partial \mathrm{PF}_{S,\partial}} \mathrm{PF}_{S,\partial} = \mathrm{PF}_{S,\partial} \coprod_{\bigcup_{p \in S} \mathcal{R}_p} \mathrm{PF}_{S,\partial} $$

(see Remark 1.2).

PF<sup>2</sup> = PF<sub>R</sub><sup>2</sup> is also the surface S (real analytic, as noted there) of [11, §3].

♦

**Remark 1.2** Per the usual correspondence (e.g. [39, Problem 11D]) between net convergence and topology Definition 1.1 gives sufficient information, but [15, §3] actually describes a local neighborhood basis of a ray r ∈ R<sub>p</sub> in PF<sub>∂</sub><sup>2</sup> = PF<sub>R,∂</sub><sup>2</sup>: the rays in R<sub>p</sub> leaning away from r at an angle < ε together with the points in H<sub>++</sub> within that ray wedge and less than ε away from p in the usual Euclidean distance.

The same source also argues that PF<sub>∂</sub><sup>2</sup> is (as the naming suggests) a topological 2-manifold (surface) with boundary, and hence its double PF is a boundary-less surface.

All of this transports immediately to the higher-dimensional, arbitrary-F setup, making

* all PF<sub>S,∂</sub><sup>n</sup> (as Definition 1.1 suggests) into contractible n-manifolds with boundary

$$ \partial \mathrm{PF}_{S, \partial}^{n} = \bigcup_{p \in S} \mathcal{R}_p $$

so that

$$ \partial \mathrm{PF}_{S, \partial}^{n} \neq \emptyset \iff S \neq \emptyset; $$

* and all doubles PF<sub>S</sub><sup>n</sup> into boundary-less n-manifolds, (path-)connected precisely when S is non-empty.

♦

[15, Proposition 3] (attributed there to Baillif) describes of the fundamental group π<sub>1</sub>(PF<sup>2</sup>) via (one version of) the *Seifert-Van Kampen theorem* [28, Theorem IV.2.2].

**Notation 1.3** Recalling (1-1), fix p, q ∈ R<sup>n-1</sup> ∼= ∂H<sub>+</sub>. In the context of Definition 1.1, define a loop α<sub>p,q</sub> lying in any PF<sub>S</sub><sup>n</sup> so long as p, q ∈ F as follows:

* if p = q then the loop is constant at the vertical ray through p = q;

* otherwise, it consists first of a path α<sub>p,q</sub><sup>+</sup> in PF<sub>S,∂</sub> from the vertical ray at p to that at q, traversing the semi-circle in H<sub>+</sub> having the segment pq as its diameter and passing through the vertical ray at p+q/2;

* followed by the reversal α<sub>p,q</sub><sup>-</sup> of that path along the mirror-image circle in the second (doubled) copy of H<sub>+</sub>.

♦

The aforementioned [15, Proposition 3], paraphrased, gives an isomorphism

<figure><img src="image_2.png" /></figure>

That argument, too, goes through with only the obvious modifications; we append a proof (in slightly different phrasing) for some semblance of completeness and the reader's convenience.
(2) The (plain, or double, or boundary-less) Prüfer n-manifold $PF_S = PF_S^n$ is the double 2$PF_{S,\partial}$ obtained as usual (e.g. [26, Example 3.80]), by gluing two copies of $PF_{S,\partial}$ along its boundary:

$$PF_S^n = PF_{S,\partial} \coprod_{\partial PF_{S,\partial}} PF_{S,\partial} = PF_{S,\partial} \coprod_{\bigcup_{p \in S} \mathcal{R}_p} PF_{S,\partial}$$

(see Remark 1.2).

$PF^2 = PF_{\mathbb{R}}^2$ is also the surface $S$ (real analytic, as noted there) of [11, §3]. ♦

**Remark 1.2** Per the usual correspondence (e.g. [39, Problem 11D]) between net convergence and topology Definition 1.1 gives sufficient information, but [15, §3] actually describes a local neighborhood basis of a ray $r \in \mathcal{R}_p$ in $PF_{\partial}^2 = PF_{\mathbb{R},\partial}^2$: the rays in $\mathcal{R}_p$ leaning away from $r$ at an angle $< \varepsilon$ together with the points in $\mathbb{H}_{++}$ within that ray wedge and less than $\varepsilon$ away from $p$ in the usual Euclidean distance.

The same source also argues that $PF_{\partial}^2$ is (as the naming suggests) a topological 2-manifold (surface) with boundary, and hence its double PF is a boundary-less surface.

All of this transports immediately to the higher-dimensional, arbitrary-$F$ setup, making

* all $PF_{S,\partial}^n$ (as Definition 1.1 suggests) into contractible $n$-manifolds with boundary

$$\partial PF_{S,\partial}^n = \bigcup_{p \in S} \mathcal{R}_p$$

so that

$$\partial PF_{S,\partial}^n \neq \emptyset \iff S \neq \emptyset;$$

* and all doubles $PF_S^n$ into boundary-less $n$-manifolds, (path-)connected precisely when $S$ is non-empty. ♦

[15, Proposition 3] (attributed there to Baillif) describes of the fundamental group $\pi_1(PF^2)$ via (one version of) the *Seifert-Van Kampen theorem* [28, Theorem IV.2.2].

**Notation 1.3** Recalling (1-1), fix $p, q \in \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$. In the context of Definition 1.1, define a loop $\alpha_{p,q}$ lying in any $PF_S^n$ so long as $p, q \in F$ as follows:

* if $p = q$ then the loop is constant at the vertical ray through $p = q$;

* otherwise, it consists first of a path $\alpha_{p,q}^+$ in $PF_{S,\partial}$ from the vertical ray at $p$ to that at $q$, traversing the semi-circle in $\mathbb{H}_+$ having the segment $\overline{pq}$ as its diameter and passing through the vertical ray at $\frac{p+q}{2}$;

* followed by the reversal $\alpha_{p,q}^-$ of that path along the mirror-image circle in the second (doubled) copy of $\mathbb{H}_+$. ♦

The aforementioned [15, Proposition 3], paraphrased, gives an isomorphism

<figure><img src="image_2.png" /></figure>

That argument, too, goes through with only the obvious modifications; we append a proof (in slightly different phrasing) for some semblance of completeness and the reader's convenience.
**Lemma 1.4** For any $n \ge 2$, non-empty $S \subseteq \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$ and $p_0 \in S$ the diagram

$$ S \setminus \{p_0\} := S^\times \xrightarrow{\text{free group } F_S^\times} \xrightarrow{\cong} \pi_1(\mathbb{P}\mathbb{F}_S^n). $$

describes the fundamental group of $\mathbb{P}\mathbb{F}_S^n$.

**Proof** Consider the two copies $\mathbb{H}_{++}^{\bullet}$, $\bullet \in \{\uparrow, \downarrow\}$ of $\mathbb{H}_{++} \subset \mathbb{R}^n$ glued along $\partial \mathbb{P}\mathbb{F}_{S,\partial}$. The open subsets

$$ U_p := \mathbb{H}_{++}^{\uparrow} \sqcup \mathcal{R}_p \sqcup \mathbb{H}_{++}^{\downarrow}, \quad p \in S. $$

are contractible, and intersect pairwise along $\mathbb{H}_{++}^{\uparrow} \sqcup \mathbb{H}_{++}^{\downarrow}$. In its groupoid-theoretic phrasing (e.g. [9, §2, Theorem]), the Van Kampen theorem identifies the groupoid $\pi_1(X, \{\uparrow, \downarrow\})$ attached ([20, Chapter 6], [8, §2]) to any choice

$$ \uparrow \in \mathbb{H}_{++}^{\uparrow} \quad \text{and} \quad \downarrow \in \mathbb{H}_{++}^{\downarrow} $$

with that of a family of segments $I_p$, $p \in S$ glued along a common pair of endpoints (and given its CW topology). After contracting one segment, this is nothing but a $S^\times$-indexed bouquet of circles, hence the conclusion [19, Example 1.21]. ■

Consider, now, for some connected $\mathbb{P}\mathbb{F} = \mathbb{P}\mathbb{F}_S^n$, the first cohomotopy group $\pi^1(\mathbb{P}\mathbb{F})$ of (0-2). In reference to Theorem 1.5, recall also the following two properties one frequently considers for functions between topological (usually at least metric) spaces:

* the *Baire class one* [24, Definition 24.1] (or *Baire one*, or *Baire-1* here, for short) are those through which the preimage of every open set is $F_\sigma$, i.e. [24, §1.A, p.1] a countable union of closed sets;

* the *Baire functions of* [36, post Example 3.1.31] (henceforth *sequentially Baire* here, for clarity, when there is danger of confusion) are the sequential pointwise limits of continuous functions.

The two properties are often equivalent and conflated, and the terminology reflects this: [32, §7], for instance, *defines* ‘first class of Baire’ as ‘(sequentially) Baire’, so does [37, Introduction], etc.

Sequentially Baire implies Baire-1 (e.g. [37, proof of Theorem 4, pp.986-987]), but the converse does not hold unconditionally [24, paragraph preceding Theorem 24.10]. The two properties are indeed equivalent for real-valued functions on metric spaces ([24, Theorem 24.10] assumes separability for the domain, but that condition does not seem necessary) or for Banach-space-valued functions on complete metric spaces [37, Theorem 4].

**Theorem 1.5** Fix a positive integer $n \ge 2$ and a non-empty subspace $S \subseteq \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$, and set $S^\times := S \setminus \{p_0\}$ for fixed $p_0 \in S$.

*The comparison embedding*

$$ \pi^1(\mathbb{P}\mathbb{F}) \xrightarrow{\text{COMP}} \text{Hom}(\pi_1(\mathbb{P}\mathbb{F}), \mathbb{Z}), \quad \mathbb{P}\mathbb{F} := \text{Prüfer n-fold } \mathbb{P}\mathbb{F}_S^n \text{ of Definition 1.1(2)} $$

identifies $\pi^1(\mathbb{P}\mathbb{F})$, as a subspace of

$$ \text{Hom}(\pi_1(\mathbb{P}\mathbb{F}), \mathbb{Z}) \cong \text{Hom}(F_{S^\times}, \mathbb{Z}) \cong (\text{FUNCTIONS } S^\times \to \mathbb{Z}), $$

with the additive group of Baire-1 functions $S^\times \to \mathbb{Z}$.
**Lemma 1.4** For any $n \ge 2$, non-empty $S \subseteq \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$ and $p_0 \in S$ the diagram

$$S \setminus \{p_0\} := S^\times \xrightarrow{\text{free group } F_S^\times} \xrightarrow{\cong} \pi_1(\mathbb{P}\mathbb{F}_S^n).$$

describes the fundamental group of $\mathbb{P}\mathbb{F}_S^n$.

**Proof** Consider the two copies $\mathbb{H}_{++}^{\bullet}$, $\bullet \in \{\uparrow, \downarrow\}$ of $\mathbb{H}_{++} \subset \mathbb{R}^n$ glued along $\partial \mathbb{P}\mathbb{F}_{S,\partial}$. The open subsets

$$U_p := \mathbb{H}_{++}^{\uparrow} \sqcup \mathcal{R}_p \sqcup \mathbb{H}_{++}^{\downarrow}, \quad p \in S.$$

are contractible, and intersect pairwise along $\mathbb{H}_{++}^{\uparrow} \sqcup \mathbb{H}_{++}^{\downarrow}$. In its groupoid-theoretic phrasing (e.g. [9, §2, Theorem]), the Van Kampen theorem identifies the groupoid $\pi_1(X, \{\uparrow, \downarrow\})$ attached ([20, Chapter 6], [8, §2]) to any choice

$$\uparrow \in \mathbb{H}_{++}^{\uparrow} \quad \text{and} \quad \downarrow \in \mathbb{H}_{++}^{\downarrow}$$

with that of a family of segments $I_p$, $p \in S$ glued along a common pair of endpoints (and given its CW topology). After contracting one segment, this is nothing but a $S^\times$-indexed bouquet of circles, hence the conclusion [19, Example 1.21]. ■

Consider, now, for some connected $\mathbb{P}\mathbb{F} = \mathbb{P}\mathbb{F}_S^n$, the first cohomotopy group $\pi^1(\mathbb{P}\mathbb{F})$ of (0-2). In reference to Theorem 1.5, recall also the following two properties one frequently considers for functions between topological (usually at least metric) spaces:

* the *Baire class one* [24, Definition 24.1] (or *Baire one*, or *Baire-1* here, for short) are those through which the preimage of every open set is $F_\sigma$, i.e. [24, §1.A, p.1] a countable union of closed sets;

* the *Baire functions of* [36, post Example 3.1.31] (henceforth *sequentially Baire* here, for clarity, when there is danger of confusion) are the sequential pointwise limits of continuous functions.

The two properties are often equivalent and conflated, and the terminology reflects this: [32, §7], for instance, *defines* ‘first class of Baire’ as ‘(sequentially) Baire’, so does [37, Introduction], etc.

Sequentially Baire implies Baire-1 (e.g. [37, proof of Theorem 4, pp.986-987]), but the converse does not hold unconditionally [24, paragraph preceding Theorem 24.10]. The two properties are indeed equivalent for real-valued functions on metric spaces ([24, Theorem 24.10] assumes separability for the domain, but that condition does not seem necessary) or for Banach-space-valued functions on complete metric spaces [37, Theorem 4].

**Theorem 1.5** Fix a positive integer $n \ge 2$ and a non-empty subspace $S \subseteq \mathbb{R}^{n-1} \cong \partial \mathbb{H}_+$, and set $S^\times := S \setminus \{p_0\}$ for fixed $p_0 \in S$.

*The comparison embedding*

$$\pi^1(\mathbb{P}\mathbb{F}) \xrightarrow{\text{COMP}} \text{Hom}(\pi_1(\mathbb{P}\mathbb{F}), \mathbb{Z}), \quad \mathbb{P}\mathbb{F} := \text{Prüfer n-fold } \mathbb{P}\mathbb{F}_S^n \text{ of Definition 1.1(2)}$$

identifies $\pi^1(\mathbb{P}\mathbb{F})$, as a subspace of

$$\text{Hom}(\pi_1(\mathbb{P}\mathbb{F}), \mathbb{Z}) \cong \text{Hom}(F_{S^\times}, \mathbb{Z}) \cong (\text{FUNCTIONS } S^\times \to \mathbb{Z}),$$

with the additive group of Baire-1 functions $S^\times \to \mathbb{Z}$.
**Proof (⇒)** The fact that the group structure on $\pi^1$ transports over to the usual additive structure on $\mathbb{Z}$-valued functions is of course not at issue; the substance of the claim is the link to the Baire property. Here, we argue that any morphism $\pi_1(\mathbb{P}F) \to \mathbb{Z}$ arising from the cohomotopy class of a continuous map $\mathbb{P}F \xrightarrow{g} \mathbb{S}^1$ is Baire-1 when restricted to the family $S^\times$ of generators $\alpha_{p_0,p}, p \in S^\times$.

$\mathbb{H}_{++}$ can be identified with the hyperbolic space $\mathbb{H}^n$ in its *upper half-plane model* [38, §2.3], whereupon the semicircles $\alpha_{p_0,p}^+$ of Notation 1.3 are precisely the geodesics connecting $p_0$ and $p$ regarded as points in the *sphere (or boundary) at infinity* ([38, §2.1], [5, Chapter II])

$$ \partial_{\infty} \mathbb{H}^n \cong \mathbb{S}^{n-1} \cong \mathbb{R}^{n-1} \cup \{\infty\} \cong \partial \mathbb{H}_+ \cup \{\infty\} $$

of $\mathbb{H}^n$. Upon effecting an automorphism of $\mathbb{H}^n$, we can assume $p_0$ itself is the ideal point $\infty \in \mathbb{S}^{-1}$, so that the half-circles $\alpha_{p_0,p}^+$ become vertical lines orthogonal to $\mathbb{R}^{n-1}$. It will be convenient to work in this parametrization, so that

* $g$ restricts to a continuous function $S^\times \times \mathbb{R}_{>0} \xrightarrow{g} \mathbb{S}^1$;

* converging to $1 \in \mathbb{S}^1$ as height increases;

* and furthermore extending continuously to the boundary $(p, 0)$ along every vertical half-line $\{p\} \times \mathbb{R}_{\ge 0}, p \in S^\times$.

The vertical path $\infty \xrightarrow{\pi^+} (p, 0)$ is $\alpha_{p_0,p}^+$, and said extension is the restriction

$$ g_p^+ := g|_{\infty \xrightarrow{\pi^+} (p,0)} $$

of $g$ to it. All of this also applies to the second copy of $\mathbb{H}_{++}$ constituting $\mathbb{P}F$: $g$, initially defined on all of $\mathbb{P}F$, also restricts the semicircle $\alpha_{p_0,p}^-$ recast as a separate (and reversed) copy $(p, 0) \xrightarrow{\pi^-} \infty$ of the path $\pi^+$, and the loop $g_*(\alpha_{p_0,p})$ in $\mathbb{S}^1$ based at $1 \in \mathbb{S}^1$ is obtained as the concatenation $g_p^+ * g_p^-$ of

$$ g_p^+ := g|_{\infty \xrightarrow{\pi^+} (p,0)} \quad \text{and} \quad g_p^- := g|_{(p,0) \xrightarrow{\pi^-} \infty}, \quad p \in S^\times. \qquad (1-2) $$

Now, $g_p^+$, regarded as a continuous map

$$ I := [I_\ell, I_r] := \overline{\infty(p, 0)} \to \mathbb{S}^1, $$

lifts to a continuous map $I \xrightarrow{h_p^+} \mathbb{R}$ with $h_p^+(I_\ell) = 0$ (by homotopy lifting [29, lemma 79.1]). The reversed $g_p^-$ similarly lifts to a map $I \xrightarrow{h_p^-} \mathbb{R}$ with $h_p^-(I_\ell) = 0$ and

$$ \text{class } [g_*(\alpha_{p_0,p})] \in \pi_1(\mathbb{S}^1) \cong \mathbb{Z} $$

is nothing but the (automatically integral) difference $h_p^-(I_r) - h_p^+(I_r)$. To conclude, simply note that because $g$ is continuous on both copies of $\mathbb{H}_{++}$ and it extends continuously along the vertical paths $\pi^\pm$ of (1-2), the maps

$$ S^\times \ni p \mapsto h_p^\pm(I_r) \in \mathbb{R} $$

are both sequential limits of continuous real-valued functions and hence [24, §24.B, 2nd paragraph] Baire-1. ■
**Proof (⇒)** The fact that the group structure on $\pi^1$ transports over to the usual additive structure on $\mathbb{Z}$-valued functions is of course not at issue; the substance of the claim is the link to the Baire property. Here, we argue that any morphism $\pi_1(\mathbb{P}F) \to \mathbb{Z}$ arising from the cohomotopy class of a continuous map $\mathbb{P}F \xrightarrow{g} \mathbb{S}^1$ is Baire-1 when restricted to the family $S^\times$ of generators $\alpha_{p_0,p}, p \in S^\times$.

$\mathbb{H}_{++}$ can be identified with the hyperbolic space $\mathbb{H}^n$ in its *upper half-plane model* [38, §2.3], whereupon the semicircles $\alpha_{p_0,p}^+$ of Notation 1.3 are precisely the geodesics connecting $p_0$ and $p$ regarded as points in the *sphere (or boundary) at infinity* ([38, §2.1], [5, Chapter II])

$$ \partial_{\infty} \mathbb{H}^n \cong \mathbb{S}^{n-1} \cong \mathbb{R}^{n-1} \cup \{\infty\} \cong \partial \mathbb{H}_+ \cup \{\infty\} $$

of $\mathbb{H}^n$. Upon effecting an automorphism of $\mathbb{H}^n$, we can assume $p_0$ itself is the ideal point $\infty \in \mathbb{S}^{-1}$, so that the half-circles $\alpha_{p_0,p}^+$ become vertical lines orthogonal to $\mathbb{R}^{n-1}$. It will be convenient to work in this parametrization, so that

* $g$ restricts to a continuous function $S^\times \times \mathbb{R}_{>0} \xrightarrow{g} \mathbb{S}^1$;

* converging to $1 \in \mathbb{S}^1$ as height increases;

* and furthermore extending continuously to the boundary $(p, 0)$ along every vertical half-line $\{p\} \times \mathbb{R}_{\ge 0}, p \in S^\times$.

The vertical path $\infty \xrightarrow{\pi^+} (p, 0)$ is $\alpha_{p_0,p}^+$, and said extension is the restriction

$$ g_p^+ := g|_{\infty \xrightarrow{\pi^+} (p,0)} $$

of $g$ to it. All of this also applies to the second copy of $\mathbb{H}_{++}$ constituting $\mathbb{P}F$: $g$, initially defined on all of $\mathbb{P}F$, also restricts the semicircle $\alpha_{p_0,p}^-$ recast as a separate (and reversed) copy $(p, 0) \xrightarrow{\pi^-} \infty$ of the path $\pi^+$, and the loop $g_*(\alpha_{p_0,p})$ in $\mathbb{S}^1$ based at $1 \in \mathbb{S}^1$ is obtained as the concatenation $g_p^+ * g_p^-$ of

$$ g_p^+ := g|_{\infty \xrightarrow{\pi^+} (p,0)} \quad \text{and} \quad g_p^- := g|_{(p,0) \xrightarrow{\pi^-} \infty}, \quad p \in S^\times. \qquad (1-2) $$

Now, $g_p^+$, regarded as a continuous map

$$ I := [I_\ell, I_r] := \overline{\infty(p, 0)} \to \mathbb{S}^1, $$

lifts to a continuous map $I \xrightarrow{h_p^+} \mathbb{R}$ with $h_p^+(I_\ell) = 0$ (by homotopy lifting [29, lemma 79.1]). The reversed $g_p^-$ similarly lifts to a map $I \xrightarrow{h_p^-} \mathbb{R}$ with $h_p^-(I_\ell) = 0$ and

$$ \text{class } [g_*(\alpha_{p_0,p})] \in \pi_1(\mathbb{S}^1) \cong \mathbb{Z} $$

is nothing but the (automatically integral) difference $h_p^-(I_r) - h_p^+(I_r)$. To conclude, simply note that because $g$ is continuous on both copies of $\mathbb{H}_{++}$ and it extends continuously along the vertical paths $\pi^\pm$ of (1-2), the maps

$$ S^\times \ni p \mapsto h_p^\pm(I_r) \in \mathbb{R} $$

are both sequential limits of continuous real-valued functions and hence [24, §24.B, 2nd paragraph] Baire-1. ■
## 2 Non-tangential limits and Borel-1 functions

Let $(B, d)$ be a metric space (frequently Polish, i.e. [24, Definition 3.1] complete and separable), and consider functions $B \xrightarrow{f} \mathbb{R}$ exhibiting two contrasting types of behavior.

* $f$ might be extensible along

$$B \cong B \times \{0\} \subset B \times \mathbb{R}_{\ge 0}$$

to a continuous function $B \times \mathbb{R}_{\ge 0} \xrightarrow{\bar{f}} \mathbb{R}$; naturally, $f$ itself must then be continuous.

* Alternatively, $f$ might be recoverable as the radial limit (in terminology borrowed from [35, §11.5], say)

$$f(b) = \lim_{t \searrow 0} \bar{f}(b, t), \quad B \times \mathbb{R}_{> 0} \xrightarrow{\text{continuous}} \mathbb{R}. \qquad (2-1)$$

It is an easy check that this is possible precisely when $f$ is a pointwise limit of a sequence of continuous functions $B \xrightarrow{f_n} \mathbb{R}$, i.e. Baire-1.

The tangential convergence of Definition 1.1(1) suggests what looks like intermediate-type behavior: $f \in \mathbb{R}^B$ might be the non-tangential limit at $t \searrow 0$ of a continuous function $\bar{f}$: in place of (2-1), impose the stronger requirement

$$f(b) = \lim_{(b',t) \xrightarrow{\text{within any } W_{b,C}} (b,0)} \bar{f}(b,t), \quad B \times \mathbb{R}_{>0} \xrightarrow{\text{continuous}} \mathbb{R}, \qquad (2-2)$$

where $W_{b,C}$, $C > 0$ is the wedge

$$W_{b,C} := \{(b', t) \in B \times \mathbb{R}_{> 0} \mid d(b, b') \le Ct\}. \qquad (2-3)$$

The language is again borrowed from complex/harmonic analysis ([35, §11.18], [40, §3], [17, §1], etc.), where one often considers such limits at points on the unit circle of functions defined in the open unit disk.

In the current setup, though, the $W_{b,C}$ of (2-3) have less to do with the geometry of tangents than with bounding, in a controlled fashion, the distance $d(b', b)$ of the first coordinate of $(b', t)$ from $b$ in terms of the second coordinate. The following gadgetry is intended to achieve that same purpose.

**Definition 2.1** (1) A (smallness) gauge is a germ at 0 of non-negative, 0-vanishing continuous functions $\mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ in the usual sheaf-theoretic sense of the term ‘germ’ ([7, §I.1, p.2], [23, Terminology 2.1.2]): an equivalence class of continuous

$$[0, \varepsilon) \xrightarrow{\phi} \mathbb{R}_{\ge 0}, \quad \phi(0) = 0$$

with two such functions declared equivalent if they are both defined and agree on some sufficiently small $[0, \varepsilon')$.

There is an obvious ordering on the collection $\mathbb{G}\mathbb{G}$ of smallness gauges: $\phi \le \phi'$ if $\phi(x) \le \phi'(x)$ for all sufficiently small $x \in \mathbb{R}_{\ge 0}$ (notation: $x \sim 0$).
## 2 Non-tangential limits and Borel-1 functions

Let $(B, d)$ be a metric space (frequently Polish, i.e. [24, Definition 3.1] complete and separable), and consider functions $B \xrightarrow{f} \mathbb{R}$ exhibiting two contrasting types of behavior.

* $f$ might be extensible along

$$B \cong B \times \{0\} \subset B \times \mathbb{R}_{\ge 0}$$

to a continuous function $B \times \mathbb{R}_{\ge 0} \xrightarrow{\bar{f}} \mathbb{R}$; naturally, $f$ itself must then be continuous.

* Alternatively, $f$ might be recoverable as the radial limit (in terminology borrowed from [35, §11.5], say)

$$f(b) = \lim_{t \searrow 0} \bar{f}(b, t), \quad B \times \mathbb{R}_{>0} \xrightarrow{\text{continuous}} \mathbb{R}. \qquad (2-1)$$

It is an easy check that this is possible precisely when $f$ is a pointwise limit of a sequence of continuous functions $B \xrightarrow{f_n} \mathbb{R}$, i.e. Baire-1.

The tangential convergence of Definition 1.1(1) suggests what looks like intermediate-type behavior: $f \in \mathbb{R}^B$ might be the non-tangential limit at $t \searrow 0$ of a continuous function $\bar{f}$: in place of (2-1), impose the stronger requirement

$$f(b) = \lim_{(b',t) \xrightarrow{\text{within any } W_{b,C}} (b,0)} \bar{f}(b,t), \quad B \times \mathbb{R}_{>0} \xrightarrow{\text{continuous}} \mathbb{R}, \qquad (2-2)$$

where $W_{b,C}$, $C > 0$ is the wedge

$$W_{b,C} := \{(b', t) \in B \times \mathbb{R}_{>0} \mid d(b, b') \le Ct\}. \qquad (2-3)$$

The language is again borrowed from complex/harmonic analysis ([35, §11.18], [40, §3], [17, §1], etc.), where one often considers such limits at points on the unit circle of functions defined in the open unit disk.

In the current setup, though, the $W_{b,C}$ of (2-3) have less to do with the geometry of tangents than with bounding, in a controlled fashion, the distance $d(b', b)$ of the first coordinate of $(b', t)$ from $b$ in terms of the second coordinate. The following gadgetry is intended to achieve that same purpose.

**Definition 2.1** (1) A (smallness) gauge is a germ at 0 of non-negative, 0-vanishing continuous functions $\mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ in the usual sheaf-theoretic sense of the term ‘germ’ ([7, §I.1, p.2], [23, Terminology 2.1.2]): an equivalence class of continuous

$$[0, \varepsilon) \xrightarrow{\phi} \mathbb{R}_{\ge 0}, \quad \phi(0) = 0$$

with two such functions declared equivalent if they are both defined and agree on some sufficiently small $[0, \varepsilon')$.

There is an obvious ordering on the collection $\mathbb{G}\mathbb{G}$ of smallness gauges: $\phi \le \phi'$ if $\phi(x) \le \phi'(x)$ for all sufficiently small $x \in \mathbb{R}_{\ge 0}$ (notation: $x \sim 0$).
(2) Let $\Phi = \{\phi\}$ be a family of gauges and $(B, d)$ a metric space. The function $B \times \mathbb{R}_{>0} \xrightarrow{\bar{f}} \mathbb{R}$ has $\Phi$-*(boundary-)limit* (or $\Phi$-*(boundary-)converges to*) a function

$$B \times \mathbb{R}_{\ge 0} \supset B \times \{0\} \cong B \xrightarrow{f} \mathbb{R}$$

if

$$f(b) = \lim_{(b',t) \xrightarrow{\text{within } W_{b,\phi}} (b,0)} \bar{f}(b',t), \quad \forall b \in B, \quad \forall \phi \in \Phi$$

for the analogue

$$W_{b,\phi} := \{(b',t) \in B \times \mathbb{R}_{>0}, t \sim 0 \mid d(b,b') \le \phi(t)\} \quad (2-4)$$

of (2-3).

(3) Similarly, a sequence $(f_n)_n$ of (typically continuous) functions $B \xrightarrow{f_n} \mathbb{R}$ is said to have $\Phi$-limit (or $\Phi$-converge to) $B \xrightarrow{f} \mathbb{R}$ if

$$f(b) = \lim_{(b', \frac{1}{n}) \xrightarrow{\text{within } W_{b,\phi}} (b,0)} f_n(b'), \quad \forall b \in B, \quad \forall \phi \in \Phi. \quad (2-5)$$

We abbreviate $\{\phi\}$-convergence to $\phi$ convergence for singletons. ♦

<figure><img src="image_3.png" /></figure>

**Theorem 2.2** For a function $B \xrightarrow{f} \mathbb{R}$ on a metric space $(B, d)$ the following conditions are equivalent.

(a) $f$ is the $\Phi$-boundary-limit of a continuous function $B \times \mathbb{R}_{>0} \xrightarrow{\bar{f}} \mathbb{R}$ for some (all) upper-bounded families $\Phi \subset (GG, \le)$.

(b) $f$ is the $\phi$-boundary-limit of a continuous function $B \times \mathbb{R}_{>0} \xrightarrow{\bar{f}} \mathbb{R}$ for some (all) $\phi \in GG$.

(c) $f$ is the $\Phi$-boundary-limit of a sequence $(f_n)$ of continuous functions for some (all) upper-bounded families $\Phi \subset (GG, \le)$.

(d) $f$ is the $\phi$-limit of a sequence $(f_n)$ of continuous functions for some (all) $\phi \in GG$.

(e) $f$ is Baire-1.

**Proof** Marking the universal ('all') and existential ('some') with '$\forall$' and '$\exists$' subscripts respectively, $\bullet_{\forall}$ formally imply the respective $\bullet_{\exists}$. Additionally,

$$(a)_{\bullet} \implies (c)_{\bullet} \quad \text{and} \quad (b)_{\bullet} \implies (d)_{\bullet}, \quad \bullet \in \{\forall, \exists\}$$

by setting $f_n := \bar{f}(\bullet, \frac{1}{n})$. Because furthermore $\phi$-convergence persists upon passing to smaller $\phi$, one can substitute an upper bound for upper-bounded $\Phi \subset GG$ to recover the 'all' family statements from their respective singleton counterparts. All conditions plainly imply (e) because pointwise convergence is simply 0-convergence, so that

$$(a)_{\forall} \iff (b)_{\forall} \implies (\text{everything else}) \implies (e). \quad (2-6)$$

As it will be convenient to ultimately settle into arguments involving the discrete conditions (c) and/or (d), we simplify matters to that extent.
(2) Let $\Phi = \{\phi\}$ be a family of gauges and $(B, d)$ a metric space. The function $B \times \mathbb{R}_{>0} \xrightarrow{\bar{f}} \mathbb{R}$ has $\Phi$-*(boundary-)limit* (or $\Phi$-*(boundary-)converges to*) a function

$$B \times \mathbb{R}_{\ge 0} \supset B \times \{0\} \cong B \xrightarrow{f} \mathbb{R}$$

if

$$f(b) = \lim_{(b',t) \xrightarrow{\text{within } W_{b,\phi}} (b,0)} \bar{f}(b',t), \quad \forall b \in B, \quad \forall \phi \in \Phi$$

for the analogue

$$W_{b,\phi} := \{(b',t) \in B \times \mathbb{R}_{>0}, t \sim 0 \mid d(b,b') \le \phi(t)\} \quad (2-4)$$

of (2-3).

(3) Similarly, a sequence $(f_n)_n$ of (typically continuous) functions $B \xrightarrow{f_n} \mathbb{R}$ is said to have $\Phi$-limit (or $\Phi$-converge to) $B \xrightarrow{f} \mathbb{R}$ if

$$f(b) = \lim_{(b', \frac{1}{n}) \xrightarrow{\text{within } W_{b,\phi}} (b,0)} f_n(b'), \quad \forall b \in B, \quad \forall \phi \in \Phi. \quad (2-5)$$

We abbreviate $\{\phi\}$-convergence to $\phi$ convergence for singletons. ♦

<figure><img src="image_3.png" /></figure>

**Theorem 2.2** For a function $B \xrightarrow{f} \mathbb{R}$ on a metric space $(B, d)$ the following conditions are equivalent.

(a) $f$ is the $\Phi$-boundary-limit of a continuous function $B \times \mathbb{R}_{>0} \xrightarrow{\bar{f}} \mathbb{R}$ for some (all) upper-bounded families $\Phi \subset (GG, \le)$.

(b) $f$ is the $\phi$-boundary-limit of a continuous function $B \times \mathbb{R}_{>0} \xrightarrow{\bar{f}} \mathbb{R}$ for some (all) $\phi \in GG$.

(c) $f$ is the $\Phi$-boundary-limit of a sequence $(f_n)$ of continuous functions for some (all) upper-bounded families $\Phi \subset (GG, \le)$.

(d) $f$ is the $\phi$-limit of a sequence $(f_n)$ of continuous functions for some (all) $\phi \in GG$.

(e) $f$ is Baire-1.

**Proof** Marking the universal ('all') and existential ('some') with '$\forall$' and '$\exists$' subscripts respectively, $\bullet_{\forall}$ formally imply the respective $\bullet_{\exists}$. Additionally,

$$(a)_{\bullet} \implies (c)_{\bullet} \quad \text{and} \quad (b)_{\bullet} \implies (d)_{\bullet}, \quad \bullet \in \{\forall, \exists\}$$

by setting $f_n := \bar{f}(\bullet, \frac{1}{n})$. Because furthermore $\phi$-convergence persists upon passing to smaller $\phi$, one can substitute an upper bound for upper-bounded $\Phi \subset GG$ to recover the 'all' family statements from their respective singleton counterparts. All conditions plainly imply (e) because pointwise convergence is simply 0-convergence, so that

$$(a)_{\forall} \iff (b)_{\forall} \implies (\text{everything else}) \implies (e). \quad (2-6)$$

As it will be convenient to ultimately settle into arguments involving the discrete conditions (c) and/or (d), we simplify matters to that extent.
(I) : $(d)_{\forall} \implies (b)_{\forall}$ Fix an arbitrary gauge $\phi$, as required by $(b)_{\forall}$. Then pick a larger gauge $\phi' \ge \phi$ with the property that

$$ \min_{I_n} \phi' > \max_{I_n} \phi, \quad \forall n \gg 0 \quad \text{with} \quad I_n := \left[ \frac{1}{n+1}, \frac{1}{n} \right], \qquad (2-7) $$

and a $\phi'$-convergent sequence $f_n \to f$ afforded by $(d)_{\forall}$. Finally, set $\bar{f}(\bullet, \frac{1}{n}) := f_n$ and thence interpolate linearly:

$$ \begin{aligned} \bar{f}\left(\bullet, \frac{s}{n+1} + \frac{1-s}{n}\right) &:= s\bar{f}\left(\bullet, \frac{1}{n+1}\right) + (1-s)\bar{f}\left(\bullet, \frac{1}{n}\right) \\ &= s f_{n+1} + (1-s)f_n, \quad s \in [0, 1]. \end{aligned} $$

By the very choice of $\phi'$, assuming $n \gg 0$ and $t \in I_n$ (in the notation of (2-7)),

$$ (b', t) \in W_{b, \phi'} \implies \left(b', \frac{1}{n+1}\right), \left(b', \frac{1}{n}\right) \in W_{b, \phi}. $$

But then for $n \gg 0$ both $f_{n+1}(b')$ and $f_n(b')$ will be close to $f(b)$, hence so will their convex combination $f(b')$.

Given (2-6), the following implication will close the circle and complete the proof.

(II) : $(e) \implies (d)_{\forall}$ Recall once more [24, Theorem 24.10]: Baire-1 real-valued functions on metric spaces are Baire (i.e. sequential limits of continuous functions); the assumed separability of the domain in loc. cit. is inessential.

In order to achieve (2-5) for such $f$ we need $|f(b) - f_n(b')|$ appropriately small. We have an estimate

$$ \begin{aligned} |f(b) - f_n(b')| &\le |f(b) - f_n(b)| + |f_n(b) - f_n(b')| \\ &\le |f_n(b) - f(b)| + \operatorname{dil}(f_n) \cdot d(b, b') \\ &\le |f_n(b) - f(b)| + \operatorname{dil}(f_n) \cdot \phi\left(\frac{1}{n}\right) \quad \text{whenever } \left(b', \frac{1}{n}\right) \in W_{b, \phi}, \end{aligned} $$

where

$$ \operatorname{dil}(\psi) := \sup_{x \neq x'} \frac{d_Y(\psi x, \psi x')}{d_X(x, x')} \in \mathbb{R}_{\ge 0} \sqcup \{\infty\}, \quad (X, d_X) \xrightarrow{\psi} (Y, d_Y) $$

denotes the dilatation ([18, Definition 1.1], [10, Definition 1.4.1]) of a function between two metric spaces. It will be enough, then, in selecting a sequence $(f_n)$ of continuous functions converging pointwise to $f$, to also ensure that

$$ \operatorname{dil}(f_n) \cdot \phi\left(\frac{1}{n}\right) \xrightarrow{n \to \infty} 0. \qquad (2-8) $$

Because $a_n := \phi(1/n)$ converges to 0, it will furthermore be enough to have $g_n$ converging pointwise to $f$ and each with finite dilatation (i.e. have $g_n$ Lipschitz [18, Definition 1.1]). For given that, we can produce the desired sequence $(f_n)$ by repeating each $g_i$ sufficiently so as to ensure the dilatations increase as slowly as (2-8) requires:

• Set the initial segment

$$ f_n = g_1, \quad 0 < n \le m_1 \quad \text{with} \quad \operatorname{dil}(g_2) \cdot a_m < 1, \quad \forall m > m_1. $$
(I) : $(d)_{\forall} \implies (b)_{\forall}$ Fix an arbitrary gauge $\phi$, as required by $(b)_{\forall}$. Then pick a larger gauge $\phi' \ge \phi$ with the property that

$$ \min_{I_n} \phi' > \max_{I_n} \phi, \quad \forall n \gg 0 \quad \text{with} \quad I_n := \left[ \frac{1}{n+1}, \frac{1}{n} \right], \qquad (2-7) $$

and a $\phi'$-convergent sequence $f_n \to f$ afforded by $(d)_{\forall}$. Finally, set $\bar{f}(\bullet, \frac{1}{n}) := f_n$ and thence interpolate linearly:

$$ \begin{aligned} \bar{f}\left(\bullet, \frac{s}{n+1} + \frac{1-s}{n}\right) &:= s\bar{f}\left(\bullet, \frac{1}{n+1}\right) + (1-s)\bar{f}\left(\bullet, \frac{1}{n}\right) \\ &= s f_{n+1} + (1-s)f_n, \quad s \in [0, 1]. \end{aligned} $$

By the very choice of $\phi'$, assuming $n \gg 0$ and $t \in I_n$ (in the notation of (2-7)),

$$ (b', t) \in W_{b, \phi'} \implies \left(b', \frac{1}{n+1}\right), \left(b', \frac{1}{n}\right) \in W_{b, \phi}. $$

But then for $n \gg 0$ both $f_{n+1}(b')$ and $f_n(b')$ will be close to $f(b)$, hence so will their convex combination $f(b')$.

Given (2-6), the following implication will close the circle and complete the proof.

(II) : $(e) \implies (d)_{\forall}$ Recall once more [24, Theorem 24.10]: Baire-1 real-valued functions on metric spaces are Baire (i.e. sequential limits of continuous functions); the assumed separability of the domain in loc. cit. is inessential.

In order to achieve (2-5) for such $f$ we need $|f(b) - f_n(b')|$ appropriately small. We have an estimate

$$ \begin{aligned} |f(b) - f_n(b')| &\le |f(b) - f_n(b)| + |f_n(b) - f_n(b')| \\ &\le |f_n(b) - f(b)| + \operatorname{dil}(f_n) \cdot d(b, b') \\ &\le |f_n(b) - f(b)| + \operatorname{dil}(f_n) \cdot \phi\left(\frac{1}{n}\right) \quad \text{whenever } \left(b', \frac{1}{n}\right) \in W_{b, \phi}, \end{aligned} $$

where

$$ \operatorname{dil}(\psi) := \sup_{x \neq x'} \frac{d_Y(\psi x, \psi x')}{d_X(x, x')} \in \mathbb{R}_{\ge 0} \sqcup \{\infty\}, \quad (X, d_X) \xrightarrow{\psi} (Y, d_Y) $$

denotes the dilatation ([18, Definition 1.1], [10, Definition 1.4.1]) of a function between two metric spaces. It will be enough, then, in selecting a sequence $(f_n)$ of continuous functions converging pointwise to $f$, to also ensure that

$$ \operatorname{dil}(f_n) \cdot \phi\left(\frac{1}{n}\right) \xrightarrow{n \to \infty} 0. \qquad (2-8) $$

Because $a_n := \phi(1/n)$ converges to 0, it will furthermore be enough to have $g_n$ converging pointwise to $f$ and each with finite dilatation (i.e. have $g_n$ Lipschitz [18, Definition 1.1]). For given that, we can produce the desired sequence $(f_n)$ by repeating each $g_i$ sufficiently so as to ensure the dilatations increase as slowly as (2-8) requires:

* Set the initial segment

$$ f_n = g_1, \quad 0 < n \le m_1 \quad \text{with} \quad \operatorname{dil}(g_2) \cdot a_m < 1, \quad \forall m > m_1. $$
• Then, similarly,

$$f_n = g_2, \quad m_1 < n \le m_2 \quad \text{with} \quad \text{dil}(g_3) \cdot a_m < \frac{1}{2}, \quad \forall m > m_2.$$

• And it should be clear how the process continues.

As for the last step of producing a Lipschitz sequence converging to $f$, this is well-known to be possible for Baire functions: it is remarked in the course of the proof of [1, Theorem 6], for instance, that this follows from [12, Proposition 3.9]. ■

**Remarks 2.3** (1) Some small elaboration might be of use to those readers who (like this one) find, on first perusal, the reference to [12, Proposition 3.9] in the proof of [1, Theorem 6] somewhat cryptic.

Consider a (real) unital linear lattice $\Phi$ of functions $X \to \mathbb{R}$ on a set $X$:

• $1 \in \Phi$ (unitality);

• and a linear space under the usual additive/scaling;

• and a lattice under the standard ordering, i.e. closed under binary (or finite) maxima and minima.

Coupled, [12, Propositions 3.1 and 3.9] then show that the class $\Phi^p$ of sequential pointwise limits of $\Phi$-members depends only on the collection

$$\mathcal{Z}(\Phi) := \{f^{-1}(0) \mid f \in \Phi\} \subseteq 2^X$$

of $\Phi$-member zero-sets. Now simply note that for metric spaces $(X, d)$ continuous and Lipschitz real-valued functions (both linear lattices containing the constants!) have the same zero-sets: precisely the closed subsets of $X$. Indeed, every closed $Z \subseteq X$ is the vanishing locus of the (1-Lipschitz) distance function

$$X \ni x \mapsto d(x, Z) := \inf\{d(x, y) \mid y \in Z\} \in \mathbb{R}_{\ge 0}$$

attached to $Z$.

(2) In fact, more is true, per the selfsame [12, Proposition 3.9]: $\Phi^p$ depends not, strictly speaking, on $\mathcal{Z}(\Phi)$, but rather only on the formally less informative collection $\mathcal{Z}(\Phi)^{c\sigma\delta}$ defined by chaining the operators

$$\text{class of sets } \mathcal{P} \mapsto \mathcal{P}^c := \{\text{complements of } Z \in \mathcal{P}\}$$

$$\mathcal{P} \mapsto \mathcal{P}^\sigma := \{\text{countable unions of } \mathcal{P}\text{-members}\} \quad \text{and}$$

$$\mathcal{P} \mapsto \mathcal{P}^\delta := \{\text{countable intersections of } \mathcal{P}\text{-members}\}.$$

Given more structure on $X$, the same principle would allow recovering Baire-1 functions as pointwise limits of even more restrictive or better-behaved classes of functions. If $X$ is a smooth manifold, for instance, the unital linear lattice of smooth functions also recovers precisely the closed subsets of $X$ as zero-sets [27, Theorem 2.29] (a result frequently attributed to Whitney, as in [22, §1, first paragraph]). ♦

For the purpose of applying all of this back to Prüfer manifolds, it will be useful to also have “relative” versions handy for the various properties listed in Theorem 2.2, all equivalent to their “absolute” counterparts.
• Then, similarly,

$$f_n = g_2, \quad m_1 < n \le m_2 \quad \text{with} \quad \text{dil}(g_3) \cdot a_m < \frac{1}{2}, \quad \forall m > m_2.$$

• And it should be clear how the process continues.

As for the last step of producing a Lipschitz sequence converging to $f$, this is well-known to be possible for Baire functions: it is remarked in the course of the proof of [1, Theorem 6], for instance, that this follows from [12, Proposition 3.9]. ■

**Remarks 2.3** (1) Some small elaboration might be of use to those readers who (like this one) find, on first perusal, the reference to [12, Proposition 3.9] in the proof of [1, Theorem 6] somewhat cryptic.

Consider a (real) unital linear lattice $\Phi$ of functions $X \to \mathbb{R}$ on a set $X$:

• $1 \in \Phi$ (unitality);

• and a linear space under the usual additive/scaling;

• and a lattice under the standard ordering, i.e. closed under binary (or finite) maxima and minima.

Coupled, [12, Propositions 3.1 and 3.9] then show that the class $\Phi^p$ of sequential pointwise limits of $\Phi$-members depends only on the collection

$$\mathcal{Z}(\Phi) := \{f^{-1}(0) \mid f \in \Phi\} \subseteq 2^X$$

of $\Phi$-member zero-sets. Now simply note that for metric spaces $(X, d)$ continuous and Lipschitz real-valued functions (both linear lattices containing the constants!) have the same zero-sets: precisely the closed subsets of $X$. Indeed, every closed $Z \subseteq X$ is the vanishing locus of the (1-Lipschitz) distance function

$$X \ni x \mapsto d(x, Z) := \inf\{d(x, y) \mid y \in Z\} \in \mathbb{R}_{\ge 0}$$

attached to $Z$.

(2) In fact, more is true, per the selfsame [12, Proposition 3.9]: $\Phi^p$ depends not, strictly speaking, on $\mathcal{Z}(\Phi)$, but rather only on the formally less informative collection $\mathcal{Z}(\Phi)^{c\sigma\delta}$ defined by chaining the operators

$$\begin{align*}
\text{class of sets } \mathcal{P} &\mapsto \mathcal{P}^c := \{\text{complements of } Z \in \mathcal{P}\} \\
\mathcal{P} &\mapsto \mathcal{P}^\sigma := \{\text{countable unions of } \mathcal{P}\text{-members}\} \quad \text{and} \\
\mathcal{P} &\mapsto \mathcal{P}^\delta := \{\text{countable intersections of } \mathcal{P}\text{-members}\}.
\end{align*}$$

Given more structure on $X$, the same principle would allow recovering Baire-1 functions as pointwise limits of even more restrictive or better-behaved classes of functions. If $X$ is a smooth manifold, for instance, the unital linear lattice of smooth functions also recovers precisely the closed subsets of $X$ as zero-sets [27, Theorem 2.29] (a result frequently attributed to Whitney, as in [22, §1, first paragraph]). ♦

For the purpose of applying all of this back to Prüfer manifolds, it will be useful to also have “relative” versions handy for the various properties listed in Theorem 2.2, all equivalent to their “absolute” counterparts.
**Definition 2.4** Consider a subset $B \stackrel{\ell}{\hookrightarrow} (B', d)$ of a metric space.

(1) A function $B \stackrel{f}{\to} \mathbb{R}$ is said to satisfy the condition (2) or (3) of Definition 2.1 $\iota$-relatively (or relatively to $\iota$) if the functions $\bar{f}$ (the sequence $(f_n)$) with the requisite properties is definable on the larger $B' \times \mathbb{R}_{>0}$ (respectively $B'$).

(2) The same terminology applies to Baire functions: $B \stackrel{f}{\to} \mathbb{R}$ is $\iota$-relatively Baire-1 if it is a sequential pointwise limit of continuous functions defined globally on $B' \supset B$. ♦

The preceding terms are convenient to have, and use, but only provisionally:

**Theorem 2.5** Let $B \stackrel{\ell}{\hookrightarrow} (B', d)$ be a subset of a metric space, correspondingly regarded as a metric space $(B', d)$ in its own right.

For a function $B \stackrel{f}{\to} \mathbb{R}$ the conditions of Theorem 2.2 are all equivalent to their $\iota$-relative counterparts in the sense of Definition 2.4.

**Proof** That every $\iota$-relative condition implies the $\iota$-relative Baire-1 property goes through as before, and the latter's equivalence to plain Baire-1 follows from [12, Proposition 3.9], as recalled in Remark 2.3(1). Indeed, although in general not all continuous functions on $B'$ extend to $B$, the two classes of functions $B' \to \mathbb{R}$ (plain continuous and extensible ones) have the same zero-sets: the closed subsets of $(B', d)$. ■

In particular, going back to the non-tangential “approach domains” (2-3):

**Corollary 2.6** A function $(B, d) \stackrel{f}{\to} \mathbb{R}$ on a metric space is Baire-1 precisely when it is $\iota$-relatively the non-tangential limit of a continuous function $B' \times \mathbb{R}_{>0} \stackrel{\bar{f}}{\to} \mathbb{R}$ in the sense of (2-2) for

* some

* or equivalently, every isometric embedding $(B, d) \stackrel{\ell}{\hookrightarrow} (B', d)$.

**Proof** This is the equivalence (e) $\iff$ (a) of Theorem 2.2, along with its relative version in Theorem 2.5, applied to the family

$$ \Phi = \{x \mapsto Cx \mid C > 0\} \subset GG $$

of smallness gauges: simply note that the family is indeed upper-bounded, for instance by (the continuous extension at 0 of) $x \mapsto \frac{1}{\ln(1/x)}$. ■

We are now ready for the outstanding implication in Theorem 1.5.

**Proof of Theorem 1.5 (⇔)** Extend a Baire-1 function $S^\times \stackrel{f}{\to} \mathbb{Z}$ to $S = S^\times \sqcup \{p_0\}$ by $p_0 \mapsto 0$ (still Baire-1), and realize that, per Corollary 2.6, as a non-tangential limit (2-1) (with $B := S$) for a continuous real-valued function $\bar{f}$ defined over $\mathbb{H}_{++}$. The imposed non-tangential-limit property then extends $\bar{f}$ across the boundary

$$ \partial \mathbb{P}F_{S,\partial} = \prod_{p \in S} \mathcal{R}_p \quad \text{of} \quad \mathbb{P}F_{S,\partial} \qquad (2-9) $$
**Definition 2.4** Consider a subset $B \stackrel{\ell}{\hookrightarrow} (B', d)$ of a metric space.

(1) A function $B \stackrel{f}{\to} \mathbb{R}$ is said to satisfy the condition (2) or (3) of Definition 2.1 $\iota$-relatively (or relatively to $\iota$) if the functions $\bar{f}$ (the sequence $(f_n)$) with the requisite properties is definable on the larger $B' \times \mathbb{R}_{>0}$ (respectively $B'$).

(2) The same terminology applies to Baire functions: $B \stackrel{f}{\to} \mathbb{R}$ is $\iota$-relatively Baire-1 if it is a sequential pointwise limit of continuous functions defined globally on $B' \supset B$. ♦

The preceding terms are convenient to have, and use, but only provisionally:

**Theorem 2.5** Let $B \stackrel{\ell}{\hookrightarrow} (B', d)$ be a subset of a metric space, correspondingly regarded as a metric space $(B', d)$ in its own right.

For a function $B \stackrel{f}{\to} \mathbb{R}$ the conditions of Theorem 2.2 are all equivalent to their $\iota$-relative counterparts in the sense of Definition 2.4.

**Proof** That every $\iota$-relative condition implies the $\iota$-relative Baire-1 property goes through as before, and the latter's equivalence to plain Baire-1 follows from [12, Proposition 3.9], as recalled in Remark 2.3(1). Indeed, although in general not all continuous functions on $B'$ extend to $B$, the two classes of functions $B' \to \mathbb{R}$ (plain continuous and extensible ones) have the same zero-sets: the closed subsets of $(B', d)$. ■

In particular, going back to the non-tangential “approach domains” (2-3):

**Corollary 2.6** A function $(B, d) \stackrel{f}{\to} \mathbb{R}$ on a metric space is Baire-1 precisely when it is $\iota$-relatively the non-tangential limit of a continuous function $B' \times \mathbb{R}_{>0} \stackrel{\bar{f}}{\to} \mathbb{R}$ in the sense of (2-2) for

* some

* or equivalently, every isometric embedding $(B, d) \stackrel{\ell}{\hookrightarrow} (B', d)$.

**Proof** This is the equivalence (e) $\iff$ (a) of Theorem 2.2, along with its relative version in Theorem 2.5, applied to the family

$$ \Phi = \{x \mapsto Cx \mid C > 0\} \subset GG $$

of smallness gauges: simply note that the family is indeed upper-bounded, for instance by (the continuous extension at 0 of) $x \mapsto \frac{1}{\ln(1/x)}$. ■

We are now ready for the outstanding implication in Theorem 1.5.

**Proof of Theorem 1.5 (⇔)** Extend a Baire-1 function $S^\times \stackrel{f}{\to} \mathbb{Z}$ to $S = S^\times \sqcup \{p_0\}$ by $p_0 \mapsto 0$ (still Baire-1), and realize that, per Corollary 2.6, as a non-tangential limit (2-1) (with $B := S$) for a continuous real-valued function $\bar{f}$ defined over $\mathbb{H}_{++}$. The imposed non-tangential-limit property then extends $\bar{f}$ across the boundary

$$ \partial \mathbb{P}F_{S,\partial} = \prod_{p \in S} \mathcal{R}_p \quad \text{of} \quad \mathbb{P}F_{S,\partial} \qquad (2-9) $$
to a continuous function (denoted economically by the same symbol) $\mathbb{P}\mathbb{F}_S \xrightarrow{\bar{f}} \mathbb{R}$, taking integer values on (2-9), and in particular collapsing every individual ray bouquet $\mathcal{R}_p$ (for every $p \in S$) to the single integer $\bar{f}(p) = f(p) \in \mathbb{Z}$.

Finally, the original $S^\times \xrightarrow{\bar{f}} \mathbb{Z}$ will be recovered as the map $\pi_1(\mathbb{P}\mathbb{F}_S) \xrightarrow{g_*} \pi_1(\mathbb{S}^1)$ for the continuous map $\mathbb{P}\mathbb{F}_S \xrightarrow{g} \mathbb{S}^1$ defined by

$$g(x) = \begin{cases} \exp(2\pi i \bar{f}(x)) \in \mathbb{S}^1 & \text{for } x \text{ in the first copy of } \mathbb{H}_{++} \\ \exp(-2\pi i \bar{f}(x)) \in \mathbb{S}^1 & \text{for } x \text{ in the second copy of } \mathbb{H}_{++} \end{cases}$$

and, naturally (and compatibly by continuity), $g(x) = 1 \in \mathbb{S}^1$ for $x$ in (2-9). ■

<figure><img src="image_4.png" /></figure>

## References

[1] Vincenzo Aversa, M. Laczkovich, and D. Preiss. Extension of differentiable functions. *Commentat. Math. Univ. Carol.*, 26:597–609, 1985. 10

[2] Mathieu Baillif. Some steps on short bridges: non-metrizable surfaces and CW-complexes. *J. Homotopy Relat. Struct.*, 7(2):153–164, 2012. 1, 3

[3] Mathieu Baillif. Collared and non-collared manifold boundaries. *Enseign. Math.*, 68(1-2):161–180, 2022. 1, 2

[4] Mathieu Baillif, Alexandre Gabard, and David Gauld. Foliations on non-metrisable manifolds: absorption by a Cantor black hole. *Proc. Am. Math. Soc.*, 142(3):1057–1069, 2014. 1

[5] Werner Ballmann. *Lectures on spaces of nonpositive curvature. With an appendix by Misha Brin: Ergodicity of geodesic flows*, volume 25 of *DMV Semin.* Basel: Birkhäuser Verlag, 1995. 6

[6] Zoltan Balogh and Gary Gruenhage. Two more perfectly normal non-metrizable manifolds. *Topology Appl.*, 151(1-3):260–272, 2005. 1

[7] Glen E. Bredon. *Sheaf theory*, volume 170 of *Grad. Texts Math.* New York, NY: Springer, 2nd ed. edition, 1997. 7

[8] R. Brown. Groupoids and van Kampen's theorem. *Proc. Lond. Math. Soc. (3)*, 17:385–401, 1967. 5

[9] Ronald Brown and Abdul Razak Salleh. A van Kampen theorem for unions of non-connected spaces. *Arch. Math.*, 42:85–88, 1984. 5

[10] Dmitri Burago, Yuri Burago, and Sergei Ivanov. *A course in metric geometry*, volume 33 of *Graduate Studies in Mathematics*. American Mathematical Society, Providence, RI, 2001. 9

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[12] Á. Császár and M. Laczkovich. Some remarks on discrete Baire classes. *Acta Math. Acad. Sci. Hung.*, 33:51–70, 1979. 10, 11
to a continuous function (denoted economically by the same symbol) $\mathbb{P}F_S \xrightarrow{\bar{f}} \mathbb{R}$, taking integer values on (2-9), and in particular collapsing every individual ray bouquet $\mathcal{R}_p$ (for every $p \in S$) to the single integer $\bar{f}(p) = f(p) \in \mathbb{Z}$.

Finally, the original $S^\times \xrightarrow{\bar{f}} \mathbb{Z}$ will be recovered as the map $\pi_1(\mathbb{P}F_S) \xrightarrow{g_*} \pi_1(\mathbb{S}^1)$ for the continuous map $\mathbb{P}F_S \xrightarrow{g} \mathbb{S}^1$ defined by

$$ g(x) = \begin{cases} \exp(2\pi i \bar{f}(x)) \in \mathbb{S}^1 & \text{for } x \text{ in the first copy of } \mathbb{H}_{++} \\ \exp(-2\pi i \bar{f}(x)) \in \mathbb{S}^1 & \text{for } x \text{ in the second copy of } \mathbb{H}_{++} \end{cases} $$

and, naturally (and compatibly by continuity), $g(x) = 1 \in \mathbb{S}^1$ for $x$ in (2-9). ■

<figure><img src="image_4.png" /></figure>

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[2] Mathieu Baillif. Some steps on short bridges: non-metrizable surfaces and CW-complexes. *J. Homotopy Relat. Struct.*, 7(2):153–164, 2012. 1, 3

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DEPARTMENT OF MATHEMATICS, UNIVERSITY AT BUFFALO
BUFFALO, NY 14260-2900, USA
*E-mail address: achirvas@buffalo.edu*
[31] Peter Nyikos. Classic problems III. *Topology Appl.*, 284:107340, 18, 2020. 1

[32] John C. Oxtoby. *Measure and category. A survey of the analogies between topological and measure spaces*. 2nd ed, volume 2 of *Grad. Texts Math*. Springer, Cham, 1980. 5

[33] T. Radó. Über den Begriff der Riemannschen Fläche. *Acta Litt. Sci. Szeged*, 2:101–121, 1925. 3

[34] Mary Ellen Rudin and Phillip Zenor. A perfectly normal nonmetrizable manifold. *Houston J. Math.*, 2:129–134, 1976. 1

[35] Walter Rudin. *Real and complex analysis*. New York, NY: McGraw-Hill, 3rd ed. edition, 1987. 7

[36] S. M. Srivastava. *A course on Borel sets*, volume 180 of *Grad. Texts Math*. New York, NY: Springer, 1998. 5

[37] Charles Stegall. Functions of the first Baire class with values in Banach spaces. *Proc. Am. Math. Soc.*, 111(4):981–991, 1991. 5

[38] William P. Thurston. Geometry and topology of three-manifolds. http://library.msri.org/books/gt3m/, 2002. accessed 2022-12-07. 6

[39] Stephen Willard. *General topology*. Dover Publications, Inc., Mineola, NY, 2004. Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581]. 1, 4

[40] Rainer Wittmann. A non-tangential limit theorem. *Osaka J. Math.*, 24:61–76, 1987. 7

DEPARTMENT OF MATHEMATICS, UNIVERSITY AT BUFFALO
BUFFALO, NY 14260-2900, USA
*E-mail address: achirvas@buffalo.edu*