A base B for a space X is said to be sharp if, whenever x ∈ X and (Bn)n∈ω is a sequence of pairwise distinct elements of B each containing x, the collection { ⋂ j≤n Bj : n ∈ ω} is a local base at x. We answer questions raised by Alleche et al. and Arhangel’skĭı et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a ...

One way to study the combinatorics of finite metric spaces is betweenness relation associated with space. In hypergraph metrization problem, one has find and characterize betweennesses collinear triples (or alternatively, non-degenerate triangles) that coincide edges a given 3-uniform hypergraph. Metrizability different kinds hypergraphs was investigated in last decades. Chen showed Steiner tri...

If σ is a topology on X and A ⊆ X, we let 〈σ,A〉 denote the topology generated by σ and A, i.e., the topology with σ ∪ {A} as a subbasis. Any refinement of a topology obtained like this – by declaring just one new set to be open – we call simple. The present paper investigates the preservation of various properties in simple refinements. The locally closed sets (sets open in their closure) play ...

We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a Gδ-diagonal, that a linearly ordered space has a point-countable bas...