Algebra ∩ topology = compactness

Compactness of Lim-inf Topology

L {inf B; B ranges over subsets of L: B ∈ F}. One can prove the following proposition (1) Let L1, L2 be complete lattices. Suppose the relational structure of L1 = the relational structure of L2. Let X1 be a non empty subset of L1, X2 be a non empty subset of L2, F1 be a filter of 2 X1 ⊆ , and F2 be a filter of 2 X2 ⊆ . If F1 = F2, then lim inf F1 = lim inf F2. Let L be a non empty FR-structure...

Compactness in Topology and Computation

Compactness in locales and in formal topology

If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale (and also for its openness). Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of ...

Lax Algebra Meets Topology

We combine two research directions of the past decade, namely the development of a lax-algebraic framework for categories of interest to topologists and analysts, and the exploration of key topological concepts, like separation and compactness, in an abstract category which comes equipped with an axiomatic notion “closed” or “proper” map. Hence, we present various candidates for such notions in...

Linear Algebra and Topology

Two real entried (respectively: orthogonal) n x n matrices^! andB are said to be linearly similar if there is an invertible real (resp. orthogonal) n x n matrix C with CAC~ = B. Of course, A, B and C may be regarded as linear functions from R (resp. isometries from S~) to itself. The matrices A and B are said to be (resp. homogeneously differentiably) differentiably similar if there is a diffeo...