Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

“…In this section, we are going to study the relationship between small loop transfer spaces at one point and these two topologies. The concept of small loop transfer space at one point was introduced in [12] as follows.…”

“…In [6, Theorem 4.12] Brodskiy et al showed that this coincidence holds at all point x ∈ X when X is a locally path connected SLT space. Moreover, Pashaei et al [12] showed that the property of being SLT at x 0 for the space X is a necessary and sufficient condition for the coincidence π wh 1 (X, x 0 ) = π qtop 1 (X, x 0 ). In the following theorem, using path Spanier groups, we give another proof for this statement.…”

“…Note that Corollary 2.6 does not hold for an SLT space at x 0 when X is not locally path connected (see [12]). It seems interesting to find a necessary and sufficient condition on X to make π wh 1 (X, x 0 ) a topological group.…”

“…In this paper, we are going to investigate on the relationship between the topological group property of π wh 1 (X, x 0 ) and π qtop 1 (X, x 0 ) and SLT spaces at one point which are introduced in [12]. In Section 2, first, we address to the relationship between path Spanier groups, introduced in [14], and SLT spaces at a point.…”

“…In Section 2, first, we address to the relationship between path Spanier groups, introduced in [14], and SLT spaces at a point. Using this relationship, we give a new proof to show that π wh 1 (X, x 0 ) = π qtop 1 (X, x 0 ) if and only if X is SLT at x 0 (see also [12]). Second, we introduce the notion of small loop transfer loop at one point (SLTL for short) and show that the property of being SLTL at x 0 for X is a necessary and sufficient condition for π wh 1 (X, x 0 ) to be a topological group.…”