On Topologized Fundamental Groups with Small Loop Transfer Viewpoints
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On Topologized Fundamental Groups with Small Loop Transfer Viewpoints Noorollah Jamali1 · Behrooz Mashayekhy2 Hamid Torabi1 · Seyyed Zeynal Pashaei1 · Mehdi Abdullahi Rashid1
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Received: 8 August 2017 / Revised: 13 January 2018 / Accepted: 30 January 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology qtop and the whisker topology, π1 (X, x0 ) and π1wh (X, x0 ), respectively. In particular, we give necessary or sufficient conditions for equality and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the converse of some of these implications do not hold, in general. Keywords Small loop transfer space · Quasitopological fundamental group · Whisker topology · Topological group Mathematics Subject Classification (2010) 57M05 · 57M12 · 55P35 · 54H11
Behrooz Mashayekhy
[email protected] Noorollah Jamali [email protected] Hamid Torabi [email protected] Seyyed Zeynal Pashaei [email protected] Mehdi Abdullahi Rashid [email protected] 1
Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
2
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
N. Jamali et al.
1 Introduction and Motivation Let P (X) denote the space of paths in X with the compact-open topology. The compactopen topology of P (X) is generated by the subbasis sets K, U = {α ∈ P (X) | α(K) ⊂ U }, where K ⊂ [0, 1] is compact and U ⊂ X is open. For given x0 ∈ X, let P (X, x0 ) = {α ∈ P (X) | α(0) = x0 } denote the space of paths starting at x0 and (X, x0 ) = {α ∈ P (X) | α(0) = x0 = α(1)} denote the space of loops based at x0 , as two well known subspaces of P (X). is the quotient space of the path space P (X, x0 ) which is defined We recall that the set X as follows. Let us consider the equivalence relation defined on P (X, x0 ) as α1 ∼ α2 if and only if α1 (1) = α2 (1) and α1 ∗ α2−1 is nullhomotopic. The equivalence class for a path α is | β α ∗ε, ε : (I, 0) → (U, α(1))} denoted by α. Note that the sets (U, α) := {β ∈ X where U is an open neighborhood of α(1) in X. This form a basis for a topology on X, is introduced by Spanier [13] and named whisker topology by Brodskiy et topology on X wh . The pointed map p : (X wh , cx ) → (X, x0 ) defined by al. [6] which is denoted by X 0 p(α) = α(1) is continuous. Moreover, if X is path connected, then p is surjective (for more details see [13, p. 82]). induces a topolFor any pointed topological space (X, x0 ), the whisker topology on X ogy on p −1 (x0 ) and let (p −1 (x0 ))wh denote the induced topology on p −1 (x0 ). Clearly, the function f : π1 (X, x0 ) → p−1 (x0 ) defined by [α] → α is a b
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