scholarly journals On special subgroups of fundamental group

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1403-1429
Author(s):  
Zadeh Ayatollah ◽  
Fatemeh Ebrahimifar ◽  
Mohammad Mahmoodi
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Fundamental Groups ◽  
Infinite Cardinal ◽  
Homotopy Classes ◽  
Hawaiian Earring

Suppose ? is a nonzero cardinal number, I is an ideal on arc connected Topological space X, and B?I(X) is the subgroup of ?1(X) (the first fundamental group of X) generated by homotopy classes of ?_I loops. The main aim of this text is to study B?I(X)s and compare them. Most interest is in ? ? {?,c} and I ? {Pfin(X), {?}}, where Pfin(X) denotes the collection of all finite subsets of X. We denote B?{?}(X) with B?(X). We prove the following statements: for arc connected topological spaces X and Y if B?(X) is isomorphic to B?(Y) for all infinite cardinal number ?, then ?1(X) is isomorphic to ?1(Y); there are arc connected topological spaces X and Y such that ?1(X) is isomorphic to ?1(Y) but B?(X) is not isomorphic to B?(Y); for arc connected topological space X we have B?(X) ? Bc(X) ? ?1(X); for Hawaiian earring X, the sets B?(X), Bc(X), and ?1(X) are pairwise distinct. So B?(X)s and B?I(X)s will help us to classify the class of all arc connected topological spaces with isomorphic fundamental groups.

scholarly journals Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1039
Author(s):  
Susmit Bagchi
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Algebraic Topology ◽  
Quotient Space ◽  
Group Structure ◽  
Topological Spaces ◽  
Fundamental Groups ◽  
Homotopy Classes ◽  
Topological Excitations ◽  
Homotopy Decomposition

The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

Uniformities on a Product

1972 ◽  
Vol 24 (3) ◽  
pp. 379-389 ◽  
Author(s):  
Anthony W. Hager
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Uniform Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
The Real ◽  
Completely Regular

All topological spaces shall be uniformizable (completely regular Hausdorff). A uniformity on X shall be viewed as a collection μ of coverings of X, via the manner of Tukey [20] and Isbell [16], and the associated uniform space denoted μX. Given the uniformizable topological space X, we shall be concerned with compatible uniformities as follows (discussed more carefully in § 1). The fine uniformity α (finest compatible with the topology); the “cardinal reflections“ αm of α (m an infinite cardinal number) ; αc, the weak uniformity generated by the real-valued continuous functions.With μ standing, generically, for one of these uniformities, we consider the question: when is μ(X × Y) = μX × μY For μ = αℵ0 (the finest compatible precompact uniformity), the problem is equivalent to that of whenβ(X × Y) = βX × βY,β denoting Stone-Cech compactification; this is answered by the theorem of Glicksberg [9]. For μ = α, we have Isbell's generalization [16, VI1.32].

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

The Number of Closed Subsets of a Topological Space

1978 ◽  
Vol 30 (02) ◽  
pp. 301-314 ◽  
Author(s):  
R. E. Hodel
Keyword(s):  
Topological Space ◽  
Basic Problem ◽  
Cardinal Number ◽  
Metrizable Space ◽  
Infinite Cardinal ◽  
Closed Sets ◽  
Complete Answer ◽  
Metrizable Spaces

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

Extensions of Pseudometrics

1966 ◽  
Vol 18 ◽  
pp. 981-998 ◽  
Author(s):  
H. L. Shapiro
Keyword(s):  
Topological Space ◽  
Cardinal Number ◽  
Infinite Cardinal ◽  
Product Topology

If γ is an infinite cardinal number, a subset S of a topological space X is said to be Pγ-embedded in X if every γ-separable continuous pseudometric on S can be extended to a γ-separable continuous pseudometric on X. (A pseudometric d on X is γ-separable if there exists a subset G of X such that |G| ⩽ 7 and such that G is dense in X relative to the pseudometric topology A pseudometric d is continuous if d is continuous relative to the product topology on X × X.) We say that S is P-embedded in X if every continuous pseudometric on S can be extended to a continuous pseudometric on X.

The projective fundamental group of a ℤ2-shift

1995 ◽  
Vol 15 (6) ◽  
pp. 1091-1118 ◽  
Author(s):  
William Geller ◽  
James Propp
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Topological Conjugacy ◽  
Fundamental Groups ◽  
Two Dimensional ◽  
Limit Group ◽  
Long Distance ◽  
Mixing Properties ◽  
Point Data ◽  
Factor Maps

AbstractWe define a new invariant for symbolic ℤ2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the ℤ2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties of the ℤ2-action that π1; of a topological space bears to π0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of ℤ2-actions, and use it to prove non-existence of certain constant-to-one factor maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have different projective fundamental groups.

An Inequality Between Numerical Homotopy Invariants

1968 ◽  
Vol 20 ◽  
pp. 1295-1299
Author(s):  
M. J. M. Priddis
Keyword(s):  
Topological Space ◽  
Nilpotency Class ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Homotopy Invariants

In (1), Berstein and Ganea denned the nilpotency class of a based topological space. For a based topological space X we write nil X for the nilpotency class of the group ΩX in the category of based topological spaces and based homotopy classes. Hilton, in (3), defined the nilpotency class, nil class K of a based semi-simplicial (s.-s.) complex; actually, the restriction of connectedness can be removed. Hence, by using the total singular complex functor S, an invariant (nil class SX) can be defined for a based topological space X.

scholarly journals MOTION PLANNING IN SPACES WITH SMALL FUNDAMENTAL GROUPS

2010 ◽  
Vol 12 (01) ◽  
pp. 107-119 ◽  
Author(s):  
ARMINDO COSTA ◽  
MICHAEL FARBER
Keyword(s):  
Fundamental Group ◽  
Motion Planning ◽  
Cohomological Dimension ◽  
Upper Bounds ◽  
Topological Spaces ◽  
Topological Complexity ◽  
Fundamental Groups ◽  
Planning Algorithms

We establish sharp upper bounds for the topological complexity TC(X) of motion planning algorithms in topological spaces X such that the fundamental group is "small", i.e. when π1(X) is cyclic of order ≤ 3 or has small cohomological dimension.

THE FUNDAMENTAL GROUP OF THE HAWAIIAN EARRING IS NOT FREE

1992 ◽  
Vol 02 (01) ◽  
pp. 33-37 ◽  
Author(s):  
BART DE SMIT
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Free Group ◽  
Arbitrary Number ◽  
Short Proof ◽  
Single Line ◽  
Countably Infinite ◽  
Hawaiian Earring

The Hawaiian earring is a topological space which is a countably infinite union of circles, that are all tangent to a single line at the same point, and whose radii tend to zero. In this note a short proof is given of a result of J.W. Morgan and I. Morrison that describes the fundamental group of this space. It is also shown that this fundamental group is not a free group, unlike the fundamental group of a wedge of an arbitrary number of circles.

scholarly journals Application Of The Seifert-Van Kampen Theorem To Certain Sums Of Topological Spaces

1970 ◽  
Vol 35 (2) ◽  
pp. 197-201
Author(s):  
Mohd. Altab Hossain
Keyword(s):  
Fundamental Group ◽  
Key Words ◽  
Commutative Diagram ◽  
Topological Spaces ◽  
Fundamental Groups

Seifert-Van Kampen theorem for the sum X + Y and external sum X ⊕Y of topological spaces is studied and the fundamental groups of these sums have been determined. Key words: Sum; External sum; Fundamental group; Commutative diagram DOI: http://dx.doi.org/10.3329/jbas.v35i2.9424 JBAS 2011; 35(2): 197-201

Sign in / Sign up

Export Citation Format

Share Document