A topological space X is said to be generated by an ideal I if for all A ⊆ X and all x ∈ A there is E ⊆ A in I such that x ∈ E, and is said to be weakly generated by I if whenever a subset A of X contains E for every E ⊆ A with E ∈ I, then A itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Haus...

In 1969, Arhangel’skĭı proved that for every Hausdorff space X, |X| ≤ 2χ(X)L(X). This beautiful inequality solved a nearly fifty-year old question raised by Alexandroff and Urysohn. In this paper we survey a wide range of generalizations and variations of Arhangel’skĭı’s inequality. We also discuss open problems and an important legacy of the theorem: the emergence of the closure method as a fu...

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form A(x) : D2u(x) = f(x). The fine scale is given by the meshsize h whereas the coarse scale is dictated by an integrodifferential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle (DMP) for any uniformly positive definite matrix A(x) provided that the mesh is weak...