Search results for "Topological space"
showing 10 items of 54 documents
Dual attachment pairs in categorically-algebraic topology
2011
[EN] The paper is a continuation of our study on developing a new approach to (lattice-valued) topological structures, which relies on category theory and universal algebra, and which is called categorically-algebraic (catalg) topology. The new framework is used to build a topological setting, based in a catalg extension of the set-theoretic membership relation "e" called dual attachment, thereby dualizing the notion of attachment introduced by the authors earlier. Following the recent interest of the fuzzy community in topological systems of S. Vickers, we clarify completely relationships between these structures and (dual) attachment, showing that unlike the former, the latter have no inh…
A topological look at human trabecular bone tissue
2017
Bone quality is affected by trabecular architecture at microscopic level. Various abnormalities of bone tissue lead to altered strength and to an increased susceptibility to fracture, such as Osteoporosis and Osteoarthritis, two major health burdens of our society. These are two complex musculoskeletal diseases that mainly concern bone tissue. In the last twenty years, there has been a growing interest in finding an appropriate topological model for the micro-architecture of trabecular bone tissue. In particular, we prove that these models involve general topological spaces. The appropriate notion to deal with is that of CW-complex.
Group topologies coarser than the Isbell topology
2011
Abstract The Isbell, compact-open and point-open topologies on the set C ( X , R ) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α ( X ) of compact families of open subsets of a topological space X . Those α ( X ) for which addition is jointly continuous at the zero function in C α ( X , R ) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α ( X ) for which C α ( X , R ) is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if X is infraconsonant. Examples based on measure theoretic methods, t…
Fuzzy algebras as a framework for fuzzy topology
2011
The paper introduces a variety-based version of the notion of the (L,M)-fuzzy topological space of Kubiak and Sostak and embeds the respective category into a suitable modification of the category of topological systems of Vickers. The new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature, paving the way for studying the problem of interweaving algebra and topology in mathematics, which was raised by Denniston, Melton and Rodabaugh in their recent research on variable-basis topological systems over the category of locales.
Categorically algebraic topology versus universal topology
2013
This paper continues to develop the theory of categorically algebraic (catalg) topology, introduced as a common framework for the majority of the existing many-valued topological settings, to provide convenient means of interaction between different approaches. Motivated by the results of universal topology of H. Herrlich, we show that a concrete category is fibre-small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures, which is definable by topological co-axioms.
Topological systems and Artin glueing
2012
Abstract Using methods of categorical fuzzy topology, the paper shows a relation between topological systems of S. Vickers and Artin glueing of M. Artin. Inspired by the problem of interrelations between algebra and topology, we show the necessary and sufficient conditions for the category, obtained by Artin glueing along an adjoint functor, to be (co)algebraic and (co)monadic, incorporating the respective result of G. Wraith. As a result, we confirm the algebraic nature of the category of topological systems, showing that it is monadic.
From Topology to Quasi-Topology. The Complexity of the Notional Domain
2016
This article examines a fundamental metalinguistic construction of the theory of enunciative operations: the notional domain. In particular, we try to explain some elementary topological concepts on which this construction is based and we try to show the key role they play in the description of some basic linguistic operations.
Hypergraph functor and attachment
2010
Using an arbitrary variety of algebras, the paper introduces a fuzzified version of the notion of attachment in a complete lattice of Guido, to provide a common framework for the concept of hypergraph functor considered by different authors in the literature. The new notion also gives rise to a category of variable-basis topological spaces which is a proper supercategory of the respective category of Rodabaugh.
On closures of discrete sets
2018
The depth of a topological space $X$ ($g(X)$) is defined as the supremum of the cardinalities of closures of discrete subsets of $X$. Solving a problem of Mart\'inez-Ruiz, Ram\'irez-P\'aramo and Romero-Morales, we prove that the cardinal inequality $|X| \leq g(X)^{L(X) \cdot F(X)}$ holds for every Hausdorff space $X$, where $L(X)$ is the Lindel\"of number of $X$ and $F(X)$ is the supremum of the cardinalities of the free sequences in $X$.
K-theory of function rings
1990
AbstractThe ring R of continuous functions on a compact topological space Xwith values in R or C is considered. It is shown that the algebraic K-theory of such rings with coefficients in ZkZ, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from Rδ (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in ZkZ of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added.