STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

Topology on Coalgebras

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

Topology and Sobolev Spaces

with 1 ≤ p <∞. W (M,N) is equipped with the standard metric d(u, v) = ‖u− v‖W1,p . Our main concern is to determine whether or not W (M,N) is path-connected and if not what can be said about its path-connected components, i.e. its W -homotopy classes. We say that u and v are W -homotopic if there is a path u ∈ C([0, 1],W (M,N)) such that u = u and u = v. We denote by ∼p the corresponding equiva...

Topology Notes: Ordered Spaces

Consider the following properties of a binary relation ≤ on a set X: (Reflexivity) For all x ∈ X, x ≤ x. (Anti-Symmetry) For all x, y ∈ X, if x ≤ y and y ≤ x, then x = y. (Transitivity) For all x, y, z ∈ X, if x ≤ y and y ≤ z, then x ≤ z. (Totality) For all x, y ∈ X, either x ≤ y or y ≤ x. A relation which satisfies reflexivity and transitivity is called a quasi-ordering. A relation which satis...

Stone-weierstrass Theorems for the Strict Topology

1. Let X be a locally compact Hausdorff space, E a (real) locally convex, complete, linear topological space, and (C*(X, E), ß) the locally convex linear space of all bounded continuous functions on X to E topologized with the strict topology ß. When E is the real numbers we denote C*(X, E) by C*(X) as usual. When E is not the real numbers, C*(X, E) is not in general an algebra, but it is a mod...