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On semilocally simply connected spaces. (English) Zbl 1219.54028

By the statement which immediately precedes Corollary 2.5.14 in E. H. Spanier’s book “Algebraic Topology” (1995; Zbl 0810.55001) the authors understand the following assertion:
\(\clubsuit\) A based space \((X,x_0)\) is semilocally 1-connected if and only if there exists an open covering \(\mathfrak U\) of \(X\) such that \(\pi(\mathfrak U, x_0)=0\).
The group \(\pi(\mathfrak U,x_0)\) is a subgroup of the fundamental group of \((X,x_0)\) and is called the Spanier group of the covering.
It turns out that this assertion is only true for locally path connected spaces. The authors construct an example of a space which is not locally path connected for which \(\clubsuit\) is not true.
In addition the authors 1) introduce a modification of the Spanier group so that \(\clubsuit\) holds (for all spaces) with this modified Spanier group and 2) They introduce a modification of the notion of semi locally 1-connectedness, such that with this concept \(\clubsuit\) holds for all spaces.

MSC:

54D05 Connected and locally connected spaces (general aspects)
55Q05 Homotopy groups, general; sets of homotopy classes
54G15 Pathological topological spaces
57M10 Covering spaces and low-dimensional topology
55Q07 Shape groups

Citations:

Zbl 0810.55001
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References:

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