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

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 \(X\).

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:

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