Function spaces and contractive extensions in Approach Theory: The role of regularity

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.

💡 Research Summary

The paper investigates the interplay between regularity properties of convergence‑approach spaces (CA‑spaces) and the behavior of function spaces equipped with the continuous convergence structure. Starting from the classical theory of convergence spaces, where regularity is known to be equivalent to the continuity of limit operators and to the possibility of extending continuous maps, the author lifts these results to the richer setting of CA‑spaces, which combine topological convergence with an approach‑distance that quantifies how “close” a filter is to a point. Two notions of regularity are introduced: (i) regularity, defined by the requirement that every filter attains its minimal approach value at each of its limit points, and (ii) strong regularity, which adds the demand that any contractive map (i.e., a map that does not increase approach values) admits a continuous extension preserving the same minimal values. Strong regularity is strictly stronger than regularity but holds in many natural examples such as complete metric spaces and completely regular spaces.

The main technical contribution is a pair of extension theorems for function spaces. The continuous convergence structure λ_c on C(X,Y) is defined by λ_c(𝔽,g)=sup_{x∈X} λ_Y(ev_x(𝔽),g(x)), where ev_x evaluates a function at a point. The author proves that if X is regular, then λ_c makes the limit operator on C(X,Y) continuous: whenever a filter of functions converges pointwise to a function g, the limit g is automatically continuous. If X is strongly regular, the result is sharpened: every contractive map f defined on a subspace A⊂X extends uniquely to a contractive map (\bar f) on the whole of X, and this extension is continuous with respect to λ_c.

A novel concept of strictness for CA‑spaces is introduced to handle extensions from subspaces. A subspace A is strict if the approach values of filters on A coincide with those of their extensions to X. Strictness ensures that the minimal approach values are preserved when moving between the subspace and the ambient space, which is crucial for the extension arguments. Using strictness, the paper establishes that regularity (respectively strong regularity) is equivalent to the universal existence of continuous (respectively contractive) extensions from any strict subspace.

The paper also supplies concrete examples: complete metric spaces satisfy strong regularity, while completely regular topological spaces are regular but may fail strong regularity. The function space C(