A notion of continuity in discrete spaces and applications

[EN] We propose a notion of continuous path for locally finite metric spaces, taking inspiration from the recent development of A-theory for locally finite connected graphs. We use this notion of continuity to derive an analogue in Z2 of the Jordan curve theorem and to extend to a quite large class of locally finite metric spaces (containing all finite metric spaces) an inequality for the ℓp-distortion of a metric space that has been recently proved by Pierre-Nicolas Jolissaint and Alain Valette for finite connected graphs.Supported by Swiss SNF Sinergia project CRSI22-130435.Capraro, V. (2013). A notion of continuity in discrete spaces and applications. Applied General Topology. 14(1):61-72. https://doi.org/10.4995/agt.2013.1618SWORD617214

Topics in uniform continuity

This paper collects results and open problems concerning several classes of functions that generalize uniform continuity in various ways, including those metric spaces (generalizing Atsuji spaces) where all continuous functions have the property of being close to uniformly continuous

Metric characterization of connectedness for topological spaces

Connectedness, path connectedness, and uniform connectedness are well-known concepts. In the traditional presentation of these concepts there is a substantial difference between connectedness and the other two notions, namely connectedness is defined as the absence of disconnectedness, while path connectedness and uniform connectedness are defined in terms of connecting paths and connecting chains, respectively. In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. Connectedness in topological spaces can also be defined in terms of chains governed by open coverings in a manner that is more reminiscent of path connectedness. We present a unifying metric formalism for connectedness, which encompasses both connectedness of topological spaces and uniform connectedness of uniform spaces, and which further extends to a hierarchy of notions of connectedness

Bounded time computation on metric spaces and Banach spaces

We extend the framework by Kawamura and Cook for investigating computational complexity for operators occurring in analysis. This model is based on second-order complexity theory for functions on the Baire space, which is lifted to metric spaces by means of representations. Time is measured in terms of the length of the input encodings and the required output precision. We propose the notions of a complete representation and of a regular representation. We show that complete representations ensure that any computable function has a time bound. Regular representations generalize Kawamura and Cook's more restrictive notion of a second-order representation, while still guaranteeing fast computability of the length of the encodings. Applying these notions, we investigate the relationship between purely metric properties of a metric space and the existence of a representation such that the metric is computable within bounded time. We show that a bound on the running time of the metric can be straightforwardly translated into size bounds of compact subsets of the metric space. Conversely, for compact spaces and for Banach spaces we construct a family of admissible, complete, regular representations that allow for fast computation of the metric and provide short encodings. Here it is necessary to trade the time bound off against the length of encodings

Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications