Open subgroups and Pontryagin duality
... Let H be a closed subgroup of an abelian topological group G. We say that H is dually closed in G if to each g ~ G \ H there corresponds a character z ~ G ~ with ZIn---0 and z(g)+0. Next, H is said to be dually embedded in G if each continuous character of H can be extended to a continuous character ...
... Let H be a closed subgroup of an abelian topological group G. We say that H is dually closed in G if to each g ~ G \ H there corresponds a character z ~ G ~ with ZIn---0 and z(g)+0. Next, H is said to be dually embedded in G if each continuous character of H can be extended to a continuous character ...
Domain representations of topological spaces
... have also studied similar notions. Ershov’s [14] representation of the Kleene–Kreisel continuous functionals is an early example of a domain representation. Scott [23] has proposed a category of equilogical spaces. The spaces in this category can also be used to represent topological spaces, see als ...
... have also studied similar notions. Ershov’s [14] representation of the Kleene–Kreisel continuous functionals is an early example of a domain representation. Scott [23] has proposed a category of equilogical spaces. The spaces in this category can also be used to represent topological spaces, see als ...
http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf
... while the pushout of the top row is the quotient X/ ∼ obtained by identifying x ∼ σ(x) for all x ∈ X. The comparison maps at the ends are X → X/ ∼, which are surjections but not isomorphisms. If we think of sets as being discrete spaces, and we take homotopy pushouts, then from this diagram we obtai ...
... while the pushout of the top row is the quotient X/ ∼ obtained by identifying x ∼ σ(x) for all x ∈ X. The comparison maps at the ends are X → X/ ∼, which are surjections but not isomorphisms. If we think of sets as being discrete spaces, and we take homotopy pushouts, then from this diagram we obtai ...
Article
... induced fuzzy topological space studied in [7, 8] are based on Theorem 3.2 of [5], which states that “a fuzzy subset A in completely induced fuzzy topological space (X, C(T)) is open iff for each rI, the strong r-cut r(A) is regular open in the topological space (X, T)”. But, incidentally, it has ...
... induced fuzzy topological space studied in [7, 8] are based on Theorem 3.2 of [5], which states that “a fuzzy subset A in completely induced fuzzy topological space (X, C(T)) is open iff for each rI, the strong r-cut r(A) is regular open in the topological space (X, T)”. But, incidentally, it has ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.