Geodesic
Group structures of a function spaces with the set-open topology
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1440-1446.

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In this paper, we find at the properties of the family $\lambda$ which imply that the space $C(X,\mathbb{R}^{\alpha})$ — the set of all continuous mappings on a Tychonoff space $X$ to the space $\mathbb{R}^{\alpha}$ with the $\lambda$-open topology is a semitopological group (paratopological group, topological group, topological vector space and other algebraic structures) under the usual operations of addition and multiplication (and multiplication by scalars). For example, if $X=[0,\omega_1)$ and $\lambda$ is a family of $C$-compact subsets of $X$, then $C_{\lambda}(X,\mathbb{R}^{\omega})$ is a semitopological group (locally convex topological vector space, topological algebra), but $C_{\lambda}(X,\mathbb{R}^{\omega_1})$ is not semitopological group.
Keywords: set-open topology, topological group, $C$-compact subset, semitopological group, paratopological group, topological vector space, $C_{\alpha}$-compact subset, topological algebra. 
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A. V. Osipov. Group structures of a function spaces with the set-open topology. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1440-1446. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a68/

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