Continuous functions in Hashimoto topologies and their algebraic properties (Q2242447)

Let \((X,\tau)\) be a \(T_1\) topological space, \(\mathcal{I}\) be an ideal of subsets of \(X\) which contains all singletons and \(\mathcal{I}\cap\tau=\{\emptyset\}\). If additionally we assume that \((X,\tau)\) is second-countable and \(\mathcal{I}\) is an \(\sigma\)-ideal then \[ \mathcal{H}=\{U\setminus I\colon U\in\tau\wedge I\in\mathcal{I}\} \] is a base of a topology called the Hashimoto topology. In the present paper the authors consider Hashimoto topologies defined for the natural topology \(\tau_{nat}\) and some \(\sigma\)-ideal \(\mathcal{I}\): \[ \mathcal{H}(\mathcal{I})=\{A\subset\mathbb{R}\colon A=U\setminus I,\textrm{ where } U\in\tau_{nat}\wedge I\in\mathcal{I}\}. \] In the first part of the paper, the authors consider the family \(\mathcal{C}_{\mathcal{H}(\mathcal{I})}\) of continuous functions \(f\colon[0,1]\to\mathbb{R}\) with the \(\mathcal{H}(\mathcal{I})\) topology on the domain and on the range of the functions. They prove that none of these families is closed under addition. Let now \(\mathcal{I_\omega}\) be the \(\sigma\)-ideal of countable sets, \(\mathcal{N}\) the \(\sigma\)-ideal of sets of Lebesgue measure zero and \(\mathcal{K}\) the \(\sigma\)-ideal of meager sets. In the second part of the article, the authors prove that if \(f\in\mathcal{C}_{\mathcal{H}(\mathcal{I})}\) for an admissible ideal \(\mathcal{I}\) then \(f\in\mathcal{C}_{\mathcal{H}(\mathcal{K})}\). Moreover, they construct a function that belongs to \(\mathcal{C}_{\mathcal{H}(\mathcal{K})}\) but does not belong to either \(\mathcal{C}_{\mathcal{H}(\mathcal{I_\omega})}\) or \(\mathcal{C}_{\mathcal{H}(\mathcal{N})}\). In the third part of the paper, the authors move to algebrability of some families of functions. They show that the families \(\mathcal{C}_{\mathcal{H}(\mathcal{I_\omega})}\cap\mathcal{C}_{\mathcal{H}(\mathcal{N})}\), \(\mathcal{C}_{\mathcal{H}(\mathcal{N})}\setminus\mathcal{C}_{\mathcal{H}(\mathcal{I_\omega})}\), \(\mathcal{C}_{\mathcal{H}(\mathcal{K})}\setminus\left(\mathcal{C}_{\mathcal{H}(\mathcal{I_\omega})}\cup\mathcal{C}_{\mathcal{H}(\mathcal{N})}\right)\) and \(\mathcal{C}_{\mathcal{H}(\mathcal{I_\omega})}\setminus\mathcal{C}_{\mathcal{H}(\mathcal{N})}\) are strongly \(\mathfrak{c}\)-algebrable. We say that the set \(A\subset\mathbb{R}\) is microscopic if for each \(\epsilon>0\) there exists a sequence of intervals \((I_n)\) such that \(A\subset\bigcup_{n\in\mathbb{N}}I_n\) and \(\lambda(I_n)\geq\epsilon^n\) for any \(n\in\mathbb{N}\). The family \(\mathcal{M}\) of microscopic sets forms a \(\sigma\)-ideal such that \(\mathcal{I_\omega}\subset\mathcal{M}\subset\mathcal{N}\). In the last part of the paper, the authors investigate some dependencies between the family \(\mathcal{C}_{\mathcal{H}(\mathcal{M})}\) and the previously mentioned families \(\mathcal{C}_{\mathcal{H}(\mathcal{N})}\) and \(\mathcal{H}(\mathcal{I_\omega})\).