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On first countable quasitopological homotopy groups. (English) Zbl 1487.55019

Considering the \(n\)th loop space of \((X,x_0)\) with the compact-open topology and the natural surjection \(q:\Omega^n(X,x_0)\rightarrow \pi_n(X,x_0)\), the homotopy group \(\pi_n(X,x_0)\) endowed with the quotient topology induced by \(q\) becomes a group with topology. At first it was claimed by some authors that this group with topology is a topological group but it has been shown that there is a gap in that claim. The main point in this gap is that the map \(q\times q\) is not a quotient map in general. After that some authors presented some examples to show that this group with topology is not a topological group. In fact, it has been shown that this group with topology is a quasitopological group which is not a topological group in general. Thereafter it has been called quasitopological homotopy group and denoted by \({\pi}^{qto}_{n}(X,x_0)\).
This fact motivated some researchers to find some conditions under which \({\pi}^{qtop}_{n}(X,x_0)\) becomes a topological group. By now, there are many significant results in this prospect. The main results of the paper under review are some interesting ones as follows:
1. If \(X\) is second countable and \({\pi}^{qtop}_{n}(X,x_0)\) is first countable, then \({\pi}^{qtop}_{n}(X,x_0)\) is a topological group.
2. Let \(X\) be a metric space, then \({\pi}^{qtop}_{n}(X,x_0)\) is a topological group if and only if \({\pi}^{qtop}_{n}(X,x_0)\) is first countable or locally compact Hausdorff.
3. Let \((X,x)=\varprojlim (X_i,x_i)\) be the inverse limit of a countably many second countable spaces \(X_i\). If \({\pi}^{qtop}_{n}(X,x_0)\) is first countable, then \({\pi}^{qtop}_{n}(X,x_0)\) is a topological group.
4. Let \(X\) be a second countable locally path connected space and \(\pi_1(X,x_0)\) be an abelian group. If \(X\) is semilocally small generated at every point except \(x_0\), then \({\pi}^{qtop}_{1}(X,x_0)\) is a first countable topological group.

MSC:

55Q05 Homotopy groups, general; sets of homotopy classes
55Q07 Shape groups
57M05 Fundamental group, presentations, free differential calculus
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