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... the additive group of real numbers. In the category of topological spaces an automorphism would be a bijective, continuous map such that its inverse map is also continuous (not guaranteed as in the group case). Concretely, the map ψ : S 1 → S 1 where ψ(α) = α + θ for some fixed angle θ is an automor ...
... the additive group of real numbers. In the category of topological spaces an automorphism would be a bijective, continuous map such that its inverse map is also continuous (not guaranteed as in the group case). Concretely, the map ψ : S 1 → S 1 where ψ(α) = α + θ for some fixed angle θ is an automor ...
On the Moreau-Rockafellar-Robinson condition in Banach spaces
... is sufficient to ensure that the infimal convolution of the conjugates of two extendedreal-valued convex lower semi-continuous functions defined on a locally convex space is exact, and that the sub-differential of the sum of these functions is the sum of their sub-differentials. During this presenta ...
... is sufficient to ensure that the infimal convolution of the conjugates of two extendedreal-valued convex lower semi-continuous functions defined on a locally convex space is exact, and that the sub-differential of the sum of these functions is the sum of their sub-differentials. During this presenta ...
THE COARSE HAWAIIAN EARRING: A COUNTABLE SPACE WITH
... Proposition 17. H is a path-connected, locally path-connected, compact, T0 space which is not T1 . Proof. Notice that the set Dn ⊂ H is homeomorphic to the coarse circle S when equipped with the subspace topology of H. Since Dn is path-connected and w0 ∈ Dn for each n ≥ 1, it follows that H is path- ...
... Proposition 17. H is a path-connected, locally path-connected, compact, T0 space which is not T1 . Proof. Notice that the set Dn ⊂ H is homeomorphic to the coarse circle S when equipped with the subspace topology of H. Since Dn is path-connected and w0 ∈ Dn for each n ≥ 1, it follows that H is path- ...
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.