Real-valued Lipschitz functions and metric properties of functions

Abstract

The purpose of this article is to explore the very general phenomenon that a function between metric spaces has a particular metric property if and only if whenever it is followed in a composition by an arbitrary real-valued Lipschitz function, the composition has this property. The key tools we use are the Efremovič lemma [21] and a theorem of Garrido and Jaramillo [22] that says that a function h between metric spaces is Lipschitz if and only if whenever it is followed by a Lipschitz real-valued function in a composition, the composition is Lipschitz. We also present a streamlined proof of the Garrido-Jaramillo result itself, but one that still relies on their natural continuous linear operator from the Lipschitz space for the target space to the Lipschitz space for the domain. Separately, we include a highly applicable uniform closure theorem that yields the most important uniform density theorems for Lipschitz-type functions as special cases.

Introduction

Analysts have little interest in topological spaces that fail to satisfy the Hausdorff separation property. They would also prefer to work in the framework of completely regular Hausdorff spaces, also called Tychonoff spaces. For one thing, in this context, such spaces have Hausdorff compactifications; in fact a topological space can be embedded in a compact Hausdorff space if and only if it is Tychonoff. Similarly, only in this framework does one have a compatible separated diagonal uniformity, so that notions such as completeness and uniform continuity can be formulated. They subsume the locally convex topological vector spaces and the locally compact Hausdorff spaces. But perhaps most importantly, it is only in this setting that the real-valued continuous functions determine the topology.

Given a family of real-valued functions $\{f_i:i\in I\}$ defined on a nonempty set Y, the initial topology ${\tau }_{\{f_i:i\in I\}}$ on Y is the topology generated by the family of subsets $\{f_i^{-1}(V):i\in I\mathrm{\text{and}}V\mathrm{\text{is open in the real line}}\}$. This is the coarsest topology one can place on Y such that each $f_i$ is continuous. A function h from a second topological space into Y equipped with such an initial topology is continuous if and only if for each $i\in I,f_i\circ h$ is continuous [40, Theorem 8.10].

As an example, if $〈Y,||\cdot ||〉$ is a real normed linear space, the initial topology on Y determined by its continuous dual is called the weak topology on Y. This will agree with the norm topology if and only if the space is finite dimensional.

Now given a topological space $〈Y,\tau 〉$ and a family of continuous real-valued functions $\{f_i:i\in I\}$ on Y, the initial topology determined by the family is in general coarser than τ. The two topologies agree provided the family separates points from closed sets, which means that whenever A is a τ-closed subset of Y and $p\in Y\backslash A$, there exists some $f_i$ such that $f_i(p)$ lies outside of the closure of $f(A)$ [40, Corollary 8.15]. The standard definition of complete regularity [40, pp. 94-95] implies that the continuous functions from $〈Y,\tau 〉$ to $[0,1]$ separate points from closed sets, so any larger class of continuous functions will do so as well. In particular, a function h into a completely regular space $〈Y,\tau 〉$ will be continuous if and only if whenever f is continuous and real-valued, $f\circ h$ is continuous.

Now if $〈Y,\rho 〉$ is a metric space with induced metric topology ${\tau }_{\rho }$, then the family of all Lipschitz functions on Y separates points from closed sets because the family of all distance functionals $y\mapsto \rho (y,p)$ where p runs over Y already does. In particular, if $h:〈X,d〉\to 〈Y,\rho 〉$ is a function, then h is continuous if and only if whenever f is a real-valued Lipschitz function on Y, $f\circ h$ is continuous. The purpose of this article is show that most of the important classes of functions between metric spaces behave analogously, including the uniformly continuous functions, the Lipschitz functions, the locally Lipschitz functions, the functions that map Cauchy sequences to Cauchy sequences, and the class of (uniformly continuous) functions that are Lipschitz in the small as introduced by Luukkainen [33]: there exist $\delta >0$ and $\lambda >0$ such that whenever $d(x_1,x_2)<\delta$ in X, we have $\rho (h(x_1),h(x_2))\le \lambda \cdot d(x_1,x_2)$.

Evidently, a bounded Lipschitz in the small function $h:X\to Y$ with parameters δ and λ is already Lipschitz, for if the diameter of $h(X)$ is M and $d(x_1,x_2)\ge \delta$, then $\rho (h(x_1),h(x_2))\le \frac{M}{\delta }d(x_1,x_2)$. If we replace the metric ρ on the target space Y by $\tilde{\rho }=\mathrm{\text{min}}\{\rho ,1\}$, then h is Lipschitz in the small if and only if h regarded as a function from X to $〈Y,\tilde{\rho }〉$ is Lipschitz [23, p. 283]. For real-valued functions, the Lipschitz in the small functions are particularly important in that such functions are uniformly dense in the real-valued uniformly continuous functions [10], [23]. At the end of the article, we give a uniform closure result for the real-valued continuous functions that are Lipschitz in the small when restricted to a particular family of subsets, provided the family is shielded from closed sets. From this single result, we can easily identify uniformly dense subclasses of Lipschitz-type functions within the most important classes of continuous functions. Our results argue for the primacy of the Lipschitz in the small functions in the study of real-valued continuous functions on metric spaces.

Along the way, we introduce and study the class of (locally Lipschitz) functions that are Lipschitz when restricted to Bourbaki bounded subsets.