publication . Article . Preprint . 2018

Realizing spaces as path-component spaces

Banakh, Taras; Brazas, Jeremy;
Open Access
  • Published: 22 Mar 2018 Journal: Fundamenta Mathematicae, volume 248, pages 79-89 (issn: 0016-2736, eissn: 1730-6329, Copyright policy)
  • Publisher: Institute of Mathematics, Polish Academy of Sciences
Abstract
The path component space of a topological space $X$ is the quotient space $\pi_0(X)$ whose points are the path components of $X$. We show that every Tychonoff space $X$ is the path-component space of a Tychonoff space $Y$ of weight $w(Y)=w(X)$ such that the natural quotient map $Y\to \pi_0(Y)=X$ is a perfect map. Hence, many topological properties of $X$ transfer to $Y$. We apply this result to construct a compact space $X\subset \mathbb{R}^3$ for which the fundamental group $\pi_1(X,x_0)$ is an uncountable, cosmic, $k_{\omega}$-topological group but for which the canonical homomorphism $\psi:\pi_1(X,x_0)\to \check{\pi}_1(X,x_0)$ to the first shape homotopy grou...
Subjects
free text keywords: Algebra and Number Theory, Topology, Mathematics, Algebra, Mathematics - General Topology, Mathematics - Algebraic Topology, 55Q52, 58B05, 54B15, 22A05, 54C10, 54G15
27 references, page 1 of 2

[1] A. Arhangel'skii, M. Tkachenko, Topological Groups and Related Structures, Series in Pure and Applied Mathematics, Atlantis Studies in Mathematics, 2008.

[2] T. Banakh, M. Vovk, M.R. Wo´ jcik, Connected economically metrizable spaces, Fund. Math. 212 (2011), 145-173.

[3] J. Brazas, Homotopy Mapping Spaces, Ph.D. Dissertation, University of New Hampshire, 2011.

[4] J. Brazas, The topological fundamental group and free topological groups, Topology Appl. 158 (2011) 779-802.

[5] J. Brazas, The fundamental group as a topological group, Topology Appl. 160 (2013) 170-188.

[6] J. Brazas, Open subgroups of free topological groups, Fundamenta Mathematicae 226 (2014) 17-40.

[7] J. Brazas, P. Fabel, On fundamental groups with the quotient topology, J. Homotopy and Related Structures 10 (2015) 71-91.

[8] J. Calcut, J. McCarthy, Discreteness and homogeneity of the topological fundamental group, Topology Proc. 34 (2009) 339-349. [OpenAIRE]

[9] R.J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press Inc., Orlando, FL, 1986.

[10] K. Eda, K. Kawamura, The fundamental groups of one-dimensional spaces, Topology Appl. 87 (1998) 163-172. [OpenAIRE]

[11] R. Engelking, General topology, Heldermann Verlag Berlin, 1989.

[12] P. Fabel, The topological hawaiian earring group does not embed in the inverse limit of free groups, Algebraic & Geometric Topology 5 (2005) 1585-1587. [OpenAIRE]

[13] P. Fabel, Multiplication is discontinuous in the hawaiian earring group, Bull. Polish Acad. Sci. Math. 59 (2011) 77-83

[14] P. Fabel, Compactly generated quasitopological homotopy groups with discontinuous multiplication, To appear in Topology Proc.

[15] H. Fischer, D. Repovsˇ, Z. Virk, A. Zastrow, On semilocally simply connected spaces, Topology Appl. 158 (2011), no. 3, 397-408.

27 references, page 1 of 2
Powered by OpenAIRE Research Graph
Any information missing or wrong?Report an Issue