Some weaker monotone separation and basis properties
A topological space is called an \(m_1\)-space (respectively an \(m_2\)-space) if each point has a closure preserving local base (respectively a closure preserving local quasi-base). It is called an \(m_3\)-space if each point has a cushioned local pair-base. Obviously, every first countable space is an \(m_1\)-space, and every \(m_1\)-space is an \(m_2\)-space. In [ibid. 19, 65-69 (1985; Zbl 0567.54012)] \textit{M. Ito} has shown that every stratifiable \(m_1\)-space is an \(M_1\)-space. Motivated by Ito's theorem, this paper contains a systematic study of \(m_i\)-spaces. It is shown that every strongly monotonically \(T_2\)-space is an \(m_2\)-space. Every regular \(m_2\)-space is an \(m_3\)-space. However, there exist first countable Hausdorff spaces, hence Hausdorff \(m_2\)-spaces, which are not \(m_3\)-spaces. Every Hausdorff \(m_3\)-space is regular. More precisely, a topological space \(X\) is a Hausdorff \(m_3\)-space if and only if it is monotonically \(T_2\), i.e. if there exists a function \(g\) assigning to each ordered pair \((x,y)\) of distinct points of \(X\) an open neighbourhood \(g(x,y)\) of \(x\) such that (i) \(g(x,y) \cap g(y,x) = \emptyset\); (ii) if \(x \in \text{cl} \bigcup \{g(y,x) : y \in A\}\), then \(x \in \text{cl} A\). Consequently, every monotonically normal \(T_1\)-space is an \(m_3\)-space. Subspaces of \(m_3\)-spaces are \(m_3\)-spaces. Box products of Hausdorff \(m_3\)-spaces are \(m_3\)-spaces. Countable \(m_3\)-spaces are \(M_1\)-spaces.
scientific article; zbMATH DE number 896579
scientific article