Essential countability of treeable equivalence relations
... is a Polish space. Then the function e : C(X, Y ) × X → Y given by e(f, x) = f (x) is continuous. Proof. See, for example, [Kur68, §IV.44.II]. Proposition 1.2. Suppose that X is a locally compact Polish space and Y is a Polish space. Then C(X, Y ) is a Polish space. Proof. See, for example, [Kur68, ...
... is a Polish space. Then the function e : C(X, Y ) × X → Y given by e(f, x) = f (x) is continuous. Proof. See, for example, [Kur68, §IV.44.II]. Proposition 1.2. Suppose that X is a locally compact Polish space and Y is a Polish space. Then C(X, Y ) is a Polish space. Proof. See, for example, [Kur68, ...
DECOMPOSITION OF CONTINUITY AND COMPLETE CONTINUITY
... (IT (γ, 12 ), 12 ), 12 ). Hence γ is 12 -fuzzy β-closed. Thus γ is 12 -fuzzy β-regular.Here γ = λ1 ∧ γ where T (λ1 ) ≥ r and γ is 12 -fuzzy β-regular. Hence γ is a 21 -weak fuzzy AB-set. Also, IT (γ, 21 ) = IT (CT (IT (γ, 21 ), 21 ), 12 ). Hence γ is a 12 -fuzzy α∗ set and thus it is a 12 -fuzzy C-s ...
... (IT (γ, 12 ), 12 ), 12 ). Hence γ is 12 -fuzzy β-closed. Thus γ is 12 -fuzzy β-regular.Here γ = λ1 ∧ γ where T (λ1 ) ≥ r and γ is 12 -fuzzy β-regular. Hence γ is a 21 -weak fuzzy AB-set. Also, IT (γ, 21 ) = IT (CT (IT (γ, 21 ), 21 ), 12 ). Hence γ is a 12 -fuzzy α∗ set and thus it is a 12 -fuzzy C-s ...
2 - Ohio State Department of Mathematics
... By work of Freedman and Casson nontriangulable manifolds exist in dimension 4 (cf Akbulut and McCarthy [2]). First, Freedman [8] showed that any homology 3– sphere bounds a contractible (topological) 4–manifold. One defines the E8 –homology manifold X 4 as follows. Start with the plumbing Q(E8 ) def ...
... By work of Freedman and Casson nontriangulable manifolds exist in dimension 4 (cf Akbulut and McCarthy [2]). First, Freedman [8] showed that any homology 3– sphere bounds a contractible (topological) 4–manifold. One defines the E8 –homology manifold X 4 as follows. Start with the plumbing Q(E8 ) def ...
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.