We present some results about the class of Alexandroff topologies (i.e. topologies where the intersection of arbitrary many open sets is open) from the perspective obtained when they are viewed as closed subsets of the Cantor cuber 2 (the power set of X with the product topology).

Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere, 1973). Combining Lawvere’s (1973) enriched-categorical and Smyth’s (1988, 1991) topological view on generalized metric spaces, it is shown how to construct (1) completion, (2) two topologies, and (3) powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ...

Working within a complete not necessarily atomic Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalize complement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Ma...

In this paper, we investigate a sheaf-theoretic interpretation of stratification learning. Motivated by the work of Alexandroff (1937) and McCord (1978), we aim to redirect efforts in the computational topology of triangulated compact polyhedra to the much more computable realm of sheaves on partially ordered sets. Our main result is the construction of stratification learning algorithms framed...

Call a topology well if and only if every open is compact. The starting point of this note is that this notion generalizes that of well-quasi order, in the sense that an Alexandroff topology is well if and only if its specialisation quasi-ordering is well. For more general topologies, this opens the way to verifying infinite transition systems based on non-well quasi ordered sets, but where the...

An Alexandroff space is a topological in which every intersection of open sets open. There one to correspondence between T0 -spaces and partially ordered (posets). We investigate -topologies on finite quandles. prove that there non-trivial topology quandle making right multiplications continuous functions if only the has more than orbit. Furthermore, we show posets quandles with n orbits are n-...

Alexandroff T0-spaces have been studied as topological models of the supports of digital images and as discrete models of continuous spaces in theoretical physics. Recently, research has been focused on the dimension of such spaces. Here we study the small inductive dimension of the digital space X (W) constructed in [15] as a minimal open quotient of a fenestrationW ofRn . There are fenestrati...

Recently W. Holliday gave a choice-free construction of canonical extension boolean algebra B as the regular open subsets Alexandroff topology on poset proper filters B. We make this point-free by replacing space with free frame $$\mathcal {L}_B$$ generated bounded meet-semilattice all (ordered reverse inclusion) and prove that booleanization is Our main result generalizes approach to category ...