FIBRED COARSE EMBEDDINGS, A-T
... if there exists • a field of Hilbert spaces {Hx }x∈X over X; • a section s : X → tx∈X Hx (i.e s(x) ∈ Hx ); • two non-decreasing functions ρ1 , ρ2 from [0, ∞) to (−∞, +∞) such that limr→∞ ρi (r) = ∞ for i = 1, 2; • A reference Hilbert space H such that for any r > 0 there exists a bounded subset Kr ⊂ ...
... if there exists • a field of Hilbert spaces {Hx }x∈X over X; • a section s : X → tx∈X Hx (i.e s(x) ∈ Hx ); • two non-decreasing functions ρ1 , ρ2 from [0, ∞) to (−∞, +∞) such that limr→∞ ρi (r) = ∞ for i = 1, 2; • A reference Hilbert space H such that for any r > 0 there exists a bounded subset Kr ⊂ ...
ON QUASI-FUZZY H-CLOSED SPACE AND CONVERGENCE Yoon
... if (i) for all A, B ∈ F we have A ∩ B ∈ F, (ii) if A ⊂ B and A ∈ F, then B ∈ F, (iii) φ ∈ / F. In a fuzzy topological space, we have various concepts of prefilter convergence. In this paper we take the following convergence concept. A prefilter F in (X, δ) converges to xα (F −→ xα ) if for any open ...
... if (i) for all A, B ∈ F we have A ∩ B ∈ F, (ii) if A ⊂ B and A ∈ F, then B ∈ F, (iii) φ ∈ / F. In a fuzzy topological space, we have various concepts of prefilter convergence. In this paper we take the following convergence concept. A prefilter F in (X, δ) converges to xα (F −→ xα ) if for any open ...
Semi-crossed Products of C*-Algebras
... each continuous and proper mapping 4: S + S defines an endomorphism a of C,(S) by a(f) =fo 4, fE C,(S). It is natural to wonder how the ringtheoretic properties of the semi-crossed product 77’ X, C,(S) reflect properties of the mapping 4, and conversely. For example, what are necessary and sufficien ...
... each continuous and proper mapping 4: S + S defines an endomorphism a of C,(S) by a(f) =fo 4, fE C,(S). It is natural to wonder how the ringtheoretic properties of the semi-crossed product 77’ X, C,(S) reflect properties of the mapping 4, and conversely. For example, what are necessary and sufficien ...
STRUCTURED SINGULAR MANIFOLDS AND FACTORIZATION
... In particular, one can define topological spaces Emb(M, N ) and Sub(M, N ) of embeddings and submersions whose homotopy types reflect the smooth topology of M and N in a significant way; this is in the contrast with the space of all smooth maps Map(M, N ), the homotopy type of which is a homotopy in ...
... In particular, one can define topological spaces Emb(M, N ) and Sub(M, N ) of embeddings and submersions whose homotopy types reflect the smooth topology of M and N in a significant way; this is in the contrast with the space of all smooth maps Map(M, N ), the homotopy type of which is a homotopy in ...
Michał Jan Cukrowski, Zbigniew Pasternak
... Proof. First we will show that if f ∈ scC0 then χ(f ) = f (p). From Lemma 4 we know that χ(ω ◦ (β1 , . . . , βn )) = ω(χ(β1 ), . . . , χ(βn )) for ω ∈ εn and β1 , . . . , βn ∈ C0 . We also know that χ(βi ) = evp (βi ) = βi (p). We can write χ(ω ◦ (β1 , . . . , βn )) = ω(β1 (p), . . . , βn (p)) = ω ◦ ...
... Proof. First we will show that if f ∈ scC0 then χ(f ) = f (p). From Lemma 4 we know that χ(ω ◦ (β1 , . . . , βn )) = ω(χ(β1 ), . . . , χ(βn )) for ω ∈ εn and β1 , . . . , βn ∈ C0 . We also know that χ(βi ) = evp (βi ) = βi (p). We can write χ(ω ◦ (β1 , . . . , βn )) = ω(β1 (p), . . . , βn (p)) = ω ◦ ...
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.