A quasi-metrization theorem for hybrid topologies on the real line in \textbf{ZF} (Q6589530)

Let us, for the sake of the reader, recall the definition of the hybrid topology on the set of real numbers. Given a 4-cover \(\mathcal{A}=\{A_1,A_2,A_3,A_4\}\) of \(\mathbb{R}\), for \(x\in \mathbb{R}\), the family \(\mathcal{B}(x)\) is defined as follows:\N\[\N\mathcal{B}(x)= \begin{cases} \{(x-\epsilon,x+\epsilon): \epsilon>0\}& \text{if}\hspace{0.2cm} x\in A_1;\\\N\{\{x\}\} & \text{if}\hspace{0.2cm} x\in A_2;\\\N\{[x,x+\epsilon): \epsilon>0\}& \text{if}\hspace{0.2cm} x\in A_3;\\\N\{(x-\epsilon,x]: \epsilon>0\}& \text{if}\hspace{0.2cm} x\in A_4. \end{cases}\N\]\NThe unique topology \(\tau_\mathcal{A}\) on the real line \(\mathbb{R}\) such that, for every \(x\in \mathbb{R}\), the family \(\mathcal{B}(x)\) is a local base at \(x\) in \( \langle \mathbb{R}, \tau_\mathcal{A}\rangle\) is called the \textit{hybrid topology determined by} \(\mathcal{A}\). The topological space \(H_4(\mathcal{A}) = \langle \mathbb{R}, \tau_\mathcal{A}\rangle\) will be called the \textit{hybrid space determined by} \(\mathcal{A}\). Note that, \(H_4(\mathcal{A})\) is a \(T_3\)-space and a generalized ordered space. Notice that if \(A_2\cup A_4 = \emptyset\), the space \(H_4(\mathcal{A})\) is the Hattori space \(H(A_1)\), that is, in \(H(A_1)\), a base of neighborhoods of \( x\in A_1\) is the family of usual Euclidean open neighborhoods of \(x\), and a base of neighborhoods of \(x\in \mathbb{R}\setminus A_1\) is the family \(\{[x, x + \epsilon) : \epsilon> 0\}\).\N\NThe set-theoretic framework for this work is the Zermelo-Fraenkel system \(\mathbf{ZF}\). The main aim of this article is to give a quasi-metrization theorem for hybrids of type \(H_4(\mathcal{A})\) in the absence of the Axiom of Choice. The authors also prove that Kofner's quasi-metrization theorem is false if there exists an infinite Dedekind-finite subset of the real line. They also give a sufficient condition for \(H_4(\mathcal{A})\) to be metrizable in \(\mathbf{ZF}\). Finally, by showing that all spaces of type \(H_4(\mathcal{A})\) are both normal and completely regular in \(\mathbf{ZF}\), they strengthen Theorem 2.3 of [\textit{T. Richmond}, Appl. Gen. Topol. 24, No. 1, 157--168 (2023; Zbl 1532.54002)] asserting that every Hattori space is normal in \(\mathbf{ZFC}\).