Rings of functions in Lipschitz topology
... Let X atd,Ibe metric spaces with metrics d and d', respectively. A map f: X-Y is Lipschitz if there is I>0 such that d'(f(*),-f(y))=Ld(x,y) for all x,yCX. The smallest such I is the Lipschitz constant lip f of f. These notions make sense also for pseudometric spaces. If each point of X has a neighbo ...
... Let X atd,Ibe metric spaces with metrics d and d', respectively. A map f: X-Y is Lipschitz if there is I>0 such that d'(f(*),-f(y))=Ld(x,y) for all x,yCX. The smallest such I is the Lipschitz constant lip f of f. These notions make sense also for pseudometric spaces. If each point of X has a neighbo ...
RIGID RATIONAL HOMOTOPY THEORY AND
... above form, such that RΓrig pX{Kq » holim∆ RΓrig pX‚ {Kq as Frobenius cdga’s. (3) Show that if A˚,‚ is a cosimplicial cdga, each with Frobenius structure, and each A˚,n admits a weight filtration, then (an explicit model for) holim∆ pA˚,‚ q admits a weight filtration. The first step was essentially ...
... above form, such that RΓrig pX{Kq » holim∆ RΓrig pX‚ {Kq as Frobenius cdga’s. (3) Show that if A˚,‚ is a cosimplicial cdga, each with Frobenius structure, and each A˚,n admits a weight filtration, then (an explicit model for) holim∆ pA˚,‚ q admits a weight filtration. The first step was essentially ...
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.