The set-open topology. (English) Zbl 1227.54021

Osipov, A. V.

Topol. Proc. 37, 205-217 (2011).

The starting point of the reviewed paper is the consideration of the following topological spaces on the set \(C(X,Y)\) of all continuous functions defined on a Tychonoff space \(X\) and with values in a metrizable topological vector space \(Y\).
\(C_{\lambda, u} (X,Y)\) for the \(\lambda\)-topology,
\(C_{\lambda} (X,Y)\) for the \(\lambda\)-open topology,
\(C_{\lambda^*}(X,Y)\) for the \(\lambda^*\)-open topology, where \(\lambda\) is an arbitrary family of subsets of the set \(X\). The elements of the standard subbases of the \(\lambda\)-topology, \(\lambda\)-open topology and \(\lambda^*\)-open topology will be denoted as follows:
\(\langle f,F,\varepsilon\rangle=\{g\in C(X,Y):\sup\{\rho (f(x),g(x)), x\in F\}<\varepsilon\), where \(F\in \lambda\),
\([F,U]=\{f\in C(X,Y): f(F)\subseteq U \}\),
\([F,U]^*=\{ f\in C(X,Y): \overline{f(F)}\subseteq U\}\)
These topological spaces in general are different. The author solves the problems of the coincidence of the \(\lambda\)-topology and the \(\lambda\)-open topology as well as the coincidence of the \(\lambda\)-topology and the \(\lambda^*\)-open topology. More precisely using the notion of \(\mathbb R\)-compactness (a notion which is very important in the study of problems concerning the coincidence of topologies on the space of functions) and the notion of boundedness, among other results, the author proves that if \(\lambda\) is a family consisting of \(\mathbb R\)-compact (respectively bounded) subsets of \(X\) such that \(A | W \in \lambda\) for any \(A \in \lambda\) and any functionally open set \(W\) such that \(A | W\) then \(C_{\lambda}(X,Y) = C_{\lambda ,u}(X, Y)\) (respectively \(C_{\lambda^*} (X,Y) = C_{\lambda ,u}(X, Y)\)), Theorem 4.4 (Theorem 4.7). Also he proves that if \(\lambda\) is a family of sets such that \(C_{\lambda}(X,Y) = C_{\lambda ,u}(X, Y)\) (respectively \(C_{\lambda^*} (X,Y) = C_{\lambda ,u}(X, Y)\)) , then 7mm

(i)

the family \(\lambda\) consists of \(\mathbb R\)-compact sets (respectively bounded sets), Theorem 4.1 (Theorem 4.5) and

(ii)

if \(\lambda_m\) is a family maximal with respect to inclusion among all the families specifying the same \(\lambda\)-open (respectively \(\lambda^*\)-open) topology on \(C(X,Y)\) then \(A | W \in \lambda_m\) for any \(A \in \lambda_m\) and any functionally open set \(W\) such that \(A | W\), Theorem 4.3 (Theorem 4.8).

This is a nice study, a continuation and improvement of the author’s earlier work together with S. E. Nokhrin entitled [“On the coincidence of the set-open and uniform topologies”, Trudy Inst. Mat. i Mekh. UrO RAN, 15, no. 2, 177–184 (2009)]

Reviewer: Nikolaos Papanastassiou (Athens)

MSC:

54C35	Function spaces in general topology
54D30	Compactness
54D60	Realcompactness and realcompactification
54G10	\(P\)-spaces
54G99	Peculiar topological spaces
54A10	Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

Keywords:

space of continuous functions; set-open topology; weak set-open topology; topology of uniform convergence on a family of sets