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A `discrete differential manifold' we call a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides us with a convenient framework for the formulation of…

We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove that any externally computable function from a computable…

In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…

We describe and classify countable Boolean rings (which may or may not have a multiplicative identity) with finitely many distinguished ideals whose elementary theory is countably categorical. This extends the description by Macintyre and…

We analyze the possibility of defining infinite-dimensional manifolds as ringed spaces. More precisely, we consider three definitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces,…

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp.…

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence…

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…

All most all the function spaces over real or complex domains and spaces of sequences, that arise in practice as examples of normed complete linear spaces (Banach spaces), are reflexive. These Banach spaces are dual to their respective…

In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related…

We introduce three new classes of pracompact spaces, consider their basic properties and relations with other compact-like spaces.

We show that the topology of pointwise convergence on scattered spaces is compatible with the group structure of their homeomorphism group. We then establish a few topological properties of the homeomorphism group of the first uncountable…

Pierce identified 3 invariants of a compact metrisable Boolean space, derived from its Cantor-Bendixson sequence, that determine the space up to homeomorphism. For locally compact spaces we define an additional invariant, the compact rank,…

Suppose $p \geq 1$ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by classifying the computable $L^p$ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we determine the…

We consider the following variation of the Scarborough-Stone problem: Is $X^\kappa$ always countably compact whenever $X$ is separable and sequentially compact?

In this paper we introduce the concept of the rectangular metric like spaces, along with its topology and we prove some fixed point theorems under different contraction principles. We introduce the concept of modified metric-like space as…

We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…

We introduce a class of variational principles on measure spaces which are causal in the sense that they generate a relation on pairs of points, giving rise to a distinction between spacelike and timelike separation. General existence…

Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and…