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We give necessary and sufficient conditions for certain pushouts of topological spaces in the category of Cech's closure spaces to agree with their pushout in the category of topological spaces. We prove that in these two categories, the…
We investigate one-point reduction methods of finite topological spaces. These methods allow one to study homotopy theory of cell complexes by means of elementary moves of their finite models. We also introduce the notion of h-regular…
We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincar\'{e} duality complexes in which a loop of a product…
This paper defines an invariant associated to Whitehead's certain exact sequence of a simply connected CW-complex which is much more elementary - and less powerful - than the boundary invariant of Baues. Nevertheless, in good cases, it…
This article introduces descriptive cellular homology on cell complexes, which is an extension of J.H.C. Whitehead's CW topology. A main result is that a descriptive cellular complex is a topology on fibres in a fibre bundle. An application…
This article introduces strongly proximal continuous (s.p.c.) functions, strong proximal equivalence (s.p.e.) and strong connectedness. A main result is that if topological spaces $X,Y$ are endowed with compatible strong proximities and…
Given CW complexes X and Y, let map(X,Y) denote the space of continuous functions from X to Y with the compact open topology. The space map(X,Y) need not have the homotopy type of a CW complex. Here the results of an extensive investigation…
We develop the theory of CW(A)-complexes, which generalizes the classical theory of CW-complexes, keeping the geometric intuition of J.H.C. Whitehead's original theory. We obtain this way generalizations of classical results, such as…
This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair…
We consider weighted coupled cell networks, that is networks where the interactions between any two cells have an associated weight that is a real valued number. Weighted networks are ubiquitous in real-world applications. We consider a…
We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It…
Graph-based signal processing techniques have become essential for handling data in non-Euclidean spaces. However, there is a growing awareness that these graph models might need to be expanded into `higher-order' domains to effectively…
Let F_*(X, Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form X cup_{alpha}e^{k+1} and a space Y whose connectivity exceeds the dimension of the…
By a formula of Farber the topological complexity TC(X) of a (p-1)-connected, m-dimensional CW-complex X is bounded above by (2m+1)/p+1. There are also various lower estimates for TC(X) such as the nilpotency of the ring $H^*(X\times…
The usual construction of a CW-approximation is functorial up to homotopy, but it is not functorial. In this note, we construct a functorial CW-approximation. Our construction takes inclusions of subspaces into inclusions of subcomplexes,…
Cell complexes (CCs) are a higher-order network model deeply rooted in algebraic topology that has gained interest in signal processing and network science recently. However, while the processing of signals supported on CCs can be described…
We introduce the notion of a protocol, which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, where the labels are required to vary continuously. If the…
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the…
The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these…
In this paper we present the notion of smooth CW complexes given by attaching cubes on the category of diffeological spaces, and we study their smooth homotopy structures related to the homotopy extension property.