Connectivity spaces
type: Item
instance of: scholarly article
title: Connectivity spaces
zbMATH Open document ID: 1398.54039
author: Bärbel M. R. Stadler
author: Peter F. Stadler
published in: Mathematics in Computer Science
publication date: 2016-01-13T00:00:00Z
review text: The concept of connectedness -- highly used in any matter related to topology -- has been treated, in a rigorous process of axiomatization, by many researchers, with different types of approaches altogether. In this paper, an attempt is made to have a glimpse of some of them; this `review-like paper' attempts to point out the underlying resemblance amongst such approaches which have been chosen by the authors. One way to obtain connectedness via axioms has been the axiomatization of connectedness, the other way being the delineation of axiom systems describing separation. The first approach is to define a connectivity space as a pair \((X,\mathcal C)\) with \(\mathcal C\subseteq 2^X\) such that (C 0): \(\emptyset\in\mathcal C\) and (C 1): \(Z_i\in\mathcal C\) for all \(i\in I\) and \(\cap Z_i\neq\emptyset\) implies \(\cup_iZ_i\in\mathcal C\). Consequently, the concept of connected component is defined as follows: for every \(A \subseteq X\) and every \(x\in X\), the set \(A[x]=\cup\{a'\subseteq X:A'\subseteq A\), \(x\in A'\) and \(A'\in\mathcal C\}\) is called the connected component of \(x\in A\); however, a basic difference with the same concept in topological spaces is that \(\{A[x]:x\in X\}\) does not define a partition of \(A\) but only a partial partition with \(\mathcal C\subseteq 2^X\) an arbitrary collection of connected sets. A pair \((A,B)\) is termed as \(\mathcal O\)-separating, if for every connected subset \(Z\in\mathcal C\) with \(Z\subseteq A\cup B\), it holds that \(Z\cap A=\emptyset\) or \(Z\cap B=\emptyset\); the collection of all separations with respect to a given collection of connected sets \(\mathcal C\) is denoted by \(\mathfrak S_{\mathcal C}\) or simply by \(\mathfrak S\); the properties satisfied by \(\mathfrak S_{\mathcal C}\) have been delineated in terms of (S 0)--(S 2) and (SR 0)--(SR 2). In another approach to connectivity, the concept of `connectivity' opening is defined as a map \(Y\): \(X\times 2^X\to 2^X:(x,A)\to a[x]\) which sa…
reviewed by: C. K. Basu
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54-02
zbMATH DE Number: 6529620
zbMATH Keywords: generalized closure function
zbMATH Keywords: generalized topology
zbMATH Keywords: fitness landscapes
zbMATH Keywords: additivity
MaRDI profile type: Publication
full work available at URL: s11786-015-0241-1
OpenAlex ID: W3188072430
Wikidata QID: Q56267212
cites work: Q3627475
cites work: On Uniform Connectedness
cites work: Connectedness in Syntopogenous Spaces
cites work: E-zusammenhängende Räume (E-connected spaces)
cites work: Generalized open sets
cites work: \(\gamma\)-connected sets
cites work: On some concepts of weak connectedness of topological spaces
cites work: A note on generalized connectedness
cites work: Q5837865
cites work: Separation spaces
cites work: Q3286846
cites work: Q5510132
cites work: Connectedness-equivalent spaces on the line
cites work: Q5813695
cites work: Q3761273
cites work: Q3336345
cites work: Q5488255
cites work: Les frontières dialectiques
cites work: On decomposing a hypergraph into \(k\) connected sub-hypergraphs
cites work: Q4889894
cites work: Q4084463
cites work: Multiscale connected operators
cites work: Q3005333
cites work: Q4370360
cites work: New insight on digital topology.
cites work: A theoretical tour of connectivity in image processing and analysis
cites work: Discrete Topological Transformations for Image Processing
cites work: Saddles and Barrier in Landscapes of Generalized Search Operators
cites work: Cycle Decompositions and Simulated Annealing
cites work: Connectivity on complete lattices
cites work: Connectivity in Lattice-Ordered Spaces
cites work: Connected sets: Bases and metrics
cites work: On Connectivity Spaces
cites work: Idempotent block splitting on partial partitions. I: Isotone operators
cites work: Idempotent block splitting on partial partitions. II: Non-isotone operators
cites work: Ordering partial partitions for image segmentation and filtering: merging, creating and inflating blocks
cites work: Q3844853
cites work: Q4147240
cites work: The relation between quasi-proximities and topological spaces
cites work: Q4692885
cites work: On Separation and Proximity Spaces
cites work: Separation relations and quasi-proximities
cites work: Q3417477
cites work: Generalized open sets in generalized topologies
cites work: On \(b\)-connectedness and \(b\)-disconnectedness and their applications
cites work: Not every Lodato proximity is covered
cites work: On the lattice of proximities of Cech compatible with a given closure space
cites work: Q3304890
cites work: Q5191278
cites work: On Topologically Induced Generalized Proximity Relations
cites work: Q5192639
cites work: Q4130059
cites work: On topologies generated by Moisil resemblance relations
cites work: Q5809626
cites work: Q3982185
cites work: Q4469988
cites work: An Axiomatic Approach to Hyperconnectivity
DOI: 10.1007/S11786-015-0241-1