scholarly journals Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces

2011 ◽  
Vol 63 (3) ◽  
pp. 769-787 ◽  
Author(s):  
Katsuya EDA
Keyword(s):  
Fundamental Groups ◽  
Atomic Property ◽  
Hawaiian Earring ◽  
Path Connected ◽  
Connected Spaces

INFINITARY COMMUTATIVITY AND ABELIANIZATION IN FUNDAMENTAL GROUPS

2020 ◽  
pp. 1-23
Author(s):  
JEREMY BRAZAS ◽  
PATRICK GILLESPIE
Keyword(s):  
Fundamental Group ◽  
Infinite Product ◽  
Homotopy Groups ◽  
Fundamental Groups ◽  
Peano Continua ◽  
Specker Group ◽  
The Subject ◽  
Set Of Points ◽  
Path Connected ◽  
Connected Spaces

Abstract Infinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$ -operations at a point $x\in X$ . Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

Embeddings of locally compact abelian p-groups in Hawaiian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanga Bavuma ◽  
Francesco G. Russo
Keyword(s):  
Algebraic Topology ◽  
Geometric Interpretation ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hawaiian Earring ◽  
Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

scholarly journals Almost-continuous path connected spaces

1981 ◽  
Vol 4 (4) ◽  
pp. 823-825
Author(s):  
Larry L. Herrington ◽  
Paul E. Long
Keyword(s):  
Continuous Function ◽  
Connected Components ◽  
Continuous Path ◽  
Open Set ◽  
Regular Open ◽  
Path Connected ◽  
Almost Continuous ◽  
Connected Spaces

M. K. Singal and Asha Rani Singal have defined an almost-continuous functionf:X→Yto be one in which for eachx∈Xand each regular-open setVcontainingf(x), there exists an openUcontainingxsuch thatf(U)⊂V. A spaceYmay now be defined to be almost-continuous path connected if for eachy0,y1∈Ythere exists an almost-continuousf:I→Ysuch thatf(0)=y0andf(1)=y1An investigation of these spaces is made culminating in a theorem showing when the almost-continuous path connected components coincide with the usual components ofY.

scholarly journals Digital Topological Properties of an Alignment of Fixed Point Sets

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 921 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Unsolved Problem ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Digital Topology ◽  
Fundamental Groups ◽  
Topological Properties ◽  
Point Sets ◽  
Fixed Point Sets ◽  
Connected Spaces

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image ( X , k ) , let F ( X ) be the set of cardinalities of the fixed point sets of all k-continuous self-maps of ( X , k ) (see Definition 4). In this paper we call it an alignment of fixed point sets of ( X , k ) . Then we have the following unsolved problem. How many components are there in F ( X ) up to 2-connectedness? In particular, let C k n , l be a simple closed k-curve with l elements in Z n and X : = C k n , l 1 ∨ C k n , l 2 be a digital wedge of C k n , l 1 and C k n , l 2 in Z n . Then we need to explore both the number of components of F ( X ) up to digital 2-connectivity (see Definition 4) and perfectness of F ( X ) (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models C 2 n n , 4 , C 3 n − 1 n , 4 , and C k n , 6 play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that X ♯ ≥ 2 .

Localization methods in the study of the homology of virtually nilpotent groups

1992 ◽  
Vol 112 (3) ◽  
pp. 551-564 ◽  
Author(s):  
Carles Casacuberta ◽  
Manuel Castellet
Keyword(s):  
Nilpotent Groups ◽  
Fundamental Groups ◽  
Connected Spaces

In a series of papers [13, 14, 15], Hilton introduced the terminology relative group to denote a group epimorphism ∈:G↠Q, and relative space to denote a map ƒ:E→B between connected spaces inducing an epimorphism of fundamental groups. He pointed out the desirability of relativizing the theory of P-localization of nilpotent groups and spaces developed in [17], and carried out the algebraic part of this project in [14, 16]. The homotopy-theoretic part was settled by Llerena in [18, 19].

scholarly journals Finite, primitive and euclidean spaces

1988 ◽  
Vol 1 (3) ◽  
pp. 177-196 ◽  
Author(s):  
Efim Khalimsky
Keyword(s):  
Image Processing ◽  
Euclidean Space ◽  
Computer Tomography ◽  
Quotient Space ◽  
Robot Vision ◽  
Euclidean Spaces ◽  
Finite Spaces ◽  
Path Connected ◽  
Connected Spaces ◽  
Digital Spaces

Integer and digital spaces are playing a significant role in digital image processing, computer graphics, computer tomography, robot vision, and many other fields dealing with finitely or countable many objects. It is proven here that every finite T0-space is a quotient space of a subspace of some simplex, i.e. of some subspace of a Euclidean space. Thus finite and digital spaces can be considered as abstract simplicial structures of subspaces of Euclidean spaces. Primitive subspaces of finite, digital, and integer spaces are introduced. They prove to be useful in the investigation of connectedness structure, which can be represented as a poset, and also in consideration of the dimension of finite spaces. Essentially T0-spaces and finitely connected and primitively path connected spaces are discussed.

Preference orders and continuous representations

2011 ◽  
Vol 61 (1) ◽  
Author(s):  
Alessandro Caterino ◽  
Rita Ceppitelli ◽  
Ghanshyam Mehta
Keyword(s):  
Representation Theorem ◽  
Topological Spaces ◽  
Continuous Representation ◽  
Representation Theorems ◽  
Preference Orders ◽  
Separable Spaces ◽  
Path Connected ◽  
Continuous Representations ◽  
Connected Spaces

AbstractIn this paper we prove some general theorems on the existence of continuous order-preserving functions on topological spaces with a continuous preorder. We use the concepts of network and netweight to prove new continuous representation theorems and we establish our main results for topological spaces that are countable unions of subspaces. Some results in the literature on path-connected, locally connected and separably connected spaces are shown to be consequences of the general theorems proved in the paper. Finally, we prove a continuous representation theorem for hereditarily separable spaces.

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