Topological Fundamental Groups and Small Generated Coverings
-
Hamid Torabi
,
Ali Pakdaman
Abstract
This paper is devoted to study some topological properties of the SG subgroup, πsg1 (X, x), of the quasitopological fundamental group of a based space (X, x), πqtop1 (X, x), its topological properties as a subgroup of the topological fundamental group πτ1 (X, x) and its influence on the existence of universal covering of X. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups.
References
[1] ARHANGEL’SKII, A.-TKACHENKO, M.: Topological Groups and Related Structures. Atlantis Stud. Math., Atlantis Press, Paris, 2008.10.2991/978-94-91216-35-0Suche in Google Scholar
[2] BISS, D.: The topological fundamental group and generalized covering spaces, Topology Appl. 124 (2002), 355-371.10.1016/S0166-8641(01)00247-4Suche in Google Scholar
[3] BOURBAKI, N.: Elements of Mathematics, General Topology I, Addison-Wesley, London, 1966.Suche in Google Scholar
[4] BRAZAS, J.: The topological fundamental group and free topological groups, Topology Appl. 158 (2011), 779-802.10.1016/j.topol.2011.01.022Suche in Google Scholar
[5] BRAZAS, J.: The fundamental group as a topological group, Topology Appl. 160 (2013), 170-188.10.1016/j.topol.2012.10.015Suche in Google Scholar
[6] CALCUT, J. S.-MCCARTHY, J. D.: Discreteness and homogeneity of the topological fundamental group, Topology Proc. 34 (2009), 339-349.Suche in Google Scholar
[7] FABEL, P.: Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology), Bull. Pol. Acad. Sci. Math. 59 (2011), 77-83.10.4064/ba59-1-9Suche in Google Scholar
[8] FISCHER, H.-ZASTROW, A.: Generalized universal coverings and the shape group, Fund. Math. 197 (2007), 167-196.10.4064/fm197-0-7Suche in Google Scholar
[9] GHANE, H.-HAMED, Z.-MASHAYEKHY, B.-MIREBRAHIMI, H.: On topological homotopy groups of n-Hawaiian like spaces, Topology Proc. 36 (2010), 255-266.Suche in Google Scholar
[10] PAKDAMAN, A.-TORABI, H.-MASHAYEKHY, B.: On H-groups and their applications to topological fundamental group, arXiv:1009.5176v1.Suche in Google Scholar
[11] PAKDAMAN, A.-TORABI, H.-MASHAYEKHY, B.: Small loop spaces and covering theory of non-homotopically Hausdorff spaces, Topology Appl. 158 (2011), 803-809.10.1016/j.topol.2011.01.024Suche in Google Scholar
[12] TORABI, H.-PAKDAMAN, A.-MASHAYEKHY, B.: On the Spanier groups and covering and semicovering spaces, arXiv:1207.4394v1.Suche in Google Scholar
[13] SPANIER, E. H.: Algebraic Topology, McGraw-Hill, New York, 1966.10.1007/978-1-4684-9322-1_5Suche in Google Scholar
[14] VIRK, Z.: Small loop spaces, Topology Appl. 157 (2010), 451-455.10.1016/j.topol.2009.10.003Suche in Google Scholar
Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Finite Mixed Sums wih Harmonic Terms
- Packing of ℝ2 by Crosses
- On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
- Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
- Generalized Derivations on Lie Ideals and Power Values on Prime Rings
- On Monoids of Injective Partial Cofinite Selfmaps
- Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
- The Controlled Convergence Theorem for the Gap-Integral
- The Solvability of a Nonlocal Boundary Value Problem
- Oscillation Criteria for Third Order Differential Equations with Functional Arguments
- Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
- On Null Lagrangians
- Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
- Convergence of Series on Large Set of Indices
- On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
- Representation of Extendible Bilinear Forms
- Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
- Topological Fundamental Groups and Small Generated Coverings
- A Relation between two Kinds of Norms for Martingales
- Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
- Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
- Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
Artikel in diesem Heft
- Finite Mixed Sums wih Harmonic Terms
- Packing of ℝ2 by Crosses
- On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
- Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
- Generalized Derivations on Lie Ideals and Power Values on Prime Rings
- On Monoids of Injective Partial Cofinite Selfmaps
- Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
- The Controlled Convergence Theorem for the Gap-Integral
- The Solvability of a Nonlocal Boundary Value Problem
- Oscillation Criteria for Third Order Differential Equations with Functional Arguments
- Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
- On Null Lagrangians
- Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
- Convergence of Series on Large Set of Indices
- On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
- Representation of Extendible Bilinear Forms
- Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
- Topological Fundamental Groups and Small Generated Coverings
- A Relation between two Kinds of Norms for Martingales
- Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
- Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
- Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
