Topological-algebraic properties of function spaces with set-open topologies. (English) Zbl 1243.54037

Osipov, Alexander V.

Topology Appl. 159, No. 3, 800-805 (2012).

Continuing his work on the coincidence of different topologies on \(C(X)\), the space of all continuous real-valued functions on a Tychonoff space \(X\) in [Topol. Proc. 37, 205–217 (2011; Zbl 1227.54021)], the author investigates the topological-algebraic properties of \(C_\lambda(X)\), the set-open topology for which the sets [F,U]=\(\{f \in C(X): f(F) \in U\}\) for \(F \in \lambda\), \(\lambda\) is a family of non-empty subsets of \(X\) and \(U\) open in \(\mathbb{R}\), form a subbase. When \(\lambda '\) consists of all \(C\)-compact subsets of every set of \(\lambda\), (i.e., \(f(A)\) is compact in \(\mathbb{R}\) for every continuous function \(f\) on \(X\), and \(A \in \lambda\)), the topology similarly generated is denoted by \(C_{\lambda '} (X)\). Here, \(\lambda\) is a \(\pi\)-net (i.e., for any open set \(U\) in \(X\), there is \(A \in \lambda\) such that \(A \subset U\)) which is equivalent to the space \(C_{\lambda}(X)\) being Hausdorff.
Contained in a series of equivalent statements, one main result is that \(C_{\lambda}(X)\) is a topological vector space (a topological group) if and only if \(\lambda\) is a family of \(C\)-compact sets and \(C_{\lambda}(X) = C_{\lambda '}(X)\). In particular, if \(C_{\lambda}(X)\) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family \(\lambda\) generated by the base of sets of the form \(\{g \in C(x): sup_{x \in F}\{|g(x)-f(x)|<\epsilon\}\) where \(F \in \lambda\) \(\epsilon >0\). Another result uses a cardinal invariant of a topological group \(G\): \(G\) is \(\omega\)-narrow if for every neighborhood \(U\) of the identity in \(G\), there is a subset \(S\) of \(G\) such that \(|S| \leq \omega\) and \(G = \{s+u: s \in S, u \in U\}\). The author shows that a topological group \(C_{\lambda}(X)\) is \(\omega\)-narrow if and only if \(\lambda\) is a family of metrizable compact subsets of \(X\).

Reviewer: Pao-Sheng Hsu (Columbia Falls)

MSC:

54C40	Algebraic properties of function spaces in general topology
54C35	Function spaces in general topology
54D60	Realcompactness and realcompactification
54H11	Topological groups (topological aspects)
46E10	Topological linear spaces of continuous, differentiable or analytic functions
54A10	Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

Keywords:

set-open topology; weak set-open topology; \(\omega\)-narrow; \(\tau\)-narrow; topological group; C-compact subset

Citations:

Zbl 1227.54021

References:

[1]	Arens, R.; Dugundji, J., Topologies for function spaces, Pacific J. Math., 1, 5-31 (1951) · Zbl 0044.11801
[2]	Arhangelʼskii, A. V.; Tkachenko, M. G., Topological Groups and Related Structures (2008), Atlantis Press
[3]	Buchwalter, H., Parties bornées dʼun espace topologique complètment régulier, Initiation à lʼAnalyse, Sem. Choquet. 9e année, 14 (1969/1970) · Zbl 0213.39604
[4]	Engelking, R., General Topology (1986), PWN: PWN Warsaw: Mir: PWN: PWN Warsaw: Mir Moscow
[5]	Gillman, L.; Jerison, M., Rings of Continuous Functions (1960), Van Nostrand: Van Nostrand New York · Zbl 0093.30001
[6]	Kundu, S.; McCoy, R. A., Topologies between compact and uniform convergence on function spaces, Int. J. Math. Math. Sci., 16, 1, 101-110 (1993) · Zbl 0798.54019
[7]	Kundu, S.; Raha, A. B., The bounded-open topology and its relatives, Rend. Istit. Mat. Univ. Trieste, 27, 61-77 (1995) · Zbl 0870.54014
[8]	McCoy, R. A.; Ntantu, I., Topological Properties of Spaces of Continuous Functions, Lecture Notes in Math., vol. 1315 (1988), Springer-Verlag: Springer-Verlag Berlin · Zbl 0621.54011
[9]	Nokhrin, S. E.; Osipov, A. V., On the coincidence of set-open and uniform topologies, Proc. Steklov Inst. Math., Suppl. 3, 184-191 (2009) · Zbl 1235.54010
[10]	Osipov, A. V., The set-open topology, Topology Proc., 37, 205-217 (2011) · Zbl 1227.54021
[11]	Velichko, N. V., \(λ\)-Topologies on function spaces, J. Math. Sci., 131, 4, 5701-5737 (2005) · Zbl 1068.22003