A STUDY ON FUZZY LOCALLY δ- CLOSED SETS

... ) which has the finite intersection property has a nonempty intersection. Corollary 5.1 A fuzzy topological space ( X, T ) is fuzzy locally compact if and only if every family of fuzzy T-closed subsets of (X, T ) with the finite intersection property has a non empty intersection. Proposition 5.3 L ...

... ) which has the finite intersection property has a nonempty intersection. Corollary 5.1 A fuzzy topological space ( X, T ) is fuzzy locally compact if and only if every family of fuzzy T-closed subsets of (X, T ) with the finite intersection property has a non empty intersection. Proposition 5.3 L ...

Properties of Algebraic Spaces

... subscheme. It is clear that R0 = s−1 (W 0 ) = t−1 (W 0 ) is a Zariski open of R which defines an étale equivalence relation on W 0 . By Spaces, Lemma 10.2 the morphism X 0 = W 0 /R0 → X is an open immersion. Hence X 0 is an algebraic space by Spaces, Lemma 11.3. By construction |X 0 | = W , i.e., X ...

... subscheme. It is clear that R0 = s−1 (W 0 ) = t−1 (W 0 ) is a Zariski open of R which defines an étale equivalence relation on W 0 . By Spaces, Lemma 10.2 the morphism X 0 = W 0 /R0 → X is an open immersion. Hence X 0 is an algebraic space by Spaces, Lemma 11.3. By construction |X 0 | = W , i.e., X ...

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... • (Split Exactness) If A → B → C is a short exact sequence of G–C∗ -algebras, split by a G-equivariant homomorphism s : C → B, then (ι( j), ι(s)) : ι(A) ⊕ ι(C) → ι(B) is an isomorphism. Moreover ι is universal (initial) among stable, split exact functors to additive categories. Kasparov theory has m ...

... • (Split Exactness) If A → B → C is a short exact sequence of G–C∗ -algebras, split by a G-equivariant homomorphism s : C → B, then (ι( j), ι(s)) : ι(A) ⊕ ι(C) → ι(B) is an isomorphism. Moreover ι is universal (initial) among stable, split exact functors to additive categories. Kasparov theory has m ...

Formal Algebraic Spaces

... Let f : X → Y be a continuous map of topological spaces. There is a functor f∗ from the category of sheaves of topological spaces, topological groups, topological rings, topological modules, to the corresponding category of sheaves on Y which is defined by setting f∗ F(V ) = F(f −1 V ) as usual. (We ...

... Let f : X → Y be a continuous map of topological spaces. There is a functor f∗ from the category of sheaves of topological spaces, topological groups, topological rings, topological modules, to the corresponding category of sheaves on Y which is defined by setting f∗ F(V ) = F(f −1 V ) as usual. (We ...

NON-HAUSDORFF GROUPOIDS, PROPER ACTIONS AND K

... an action of a groupoid G on a space Z is proper if and only if the crossedproduct groupoid Z o G is proper). Our definition is as follows: a topological groupoid G is proper if the map (r, s) : G → G(0) × G(0) is proper in the sense of Bourbaki [5]. Properness is shown in Section 2 to be invariant ...

... an action of a groupoid G on a space Z is proper if and only if the crossedproduct groupoid Z o G is proper). Our definition is as follows: a topological groupoid G is proper if the map (r, s) : G → G(0) × G(0) is proper in the sense of Bourbaki [5]. Properness is shown in Section 2 to be invariant ...

Properties of Schemes

... Lemma 5.9. Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point. Proof. The second assertion follows from the first (using Schemes, ...

... Lemma 5.9. Any nonempty locally Noetherian scheme has a closed point. Any nonempty closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point. Proof. The second assertion follows from the first (using Schemes, ...

Algebraic Topology

... rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the i ...

... rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the i ...

Derived algebraic geometry

... Just as an ordinary scheme is defined to be “something which looks locally like Spec A where A is a commutative ring”, a derived scheme will be defined to be “something which looks locally like Spec A where A is a topological commutative ring”. Remark 1.1.3. We should emphasize that the topology of ...

... Just as an ordinary scheme is defined to be “something which looks locally like Spec A where A is a commutative ring”, a derived scheme will be defined to be “something which looks locally like Spec A where A is a topological commutative ring”. Remark 1.1.3. We should emphasize that the topology of ...

Embeddings of compact convex sets and locally compact cones

... subsets of general linear spaces can differ from the compact convex subspaces of locally convex spaces or some mild variant thereof. For example, the following is a question posed by V. Klee [8]: Does every element of a compact convex subset K of a topological vector space possess a basis of neighbo ...

... subsets of general linear spaces can differ from the compact convex subspaces of locally convex spaces or some mild variant thereof. For example, the following is a question posed by V. Klee [8]: Does every element of a compact convex subset K of a topological vector space possess a basis of neighbo ...

FIBRATIONS OF TOPOLOGICAL STACKS Contents 1. Introduction 2

... We do not know if Proposition 2.7 remains true when the maps i or j are not embeddings, or when X is an arbitrary topological stack. It appears though that, even when (Y, A) is a nice pair (say an inclusion of a finite CW complex into another), the quotient space Y /A may not in general have the uni ...

... We do not know if Proposition 2.7 remains true when the maps i or j are not embeddings, or when X is an arbitrary topological stack. It appears though that, even when (Y, A) is a nice pair (say an inclusion of a finite CW complex into another), the quotient space Y /A may not in general have the uni ...

“Research Note” TOPOLOGICAL RING

... 1. INTRODUCTION Let X be a connected topological group with zero element 0, and let p : X → X be the universal covering map of the underlying space of X. It follows easily from classical properties of lifting maps to covering = 0 , there is a structure of topological group on X such that spaces ...

... 1. INTRODUCTION Let X be a connected topological group with zero element 0, and let p : X → X be the universal covering map of the underlying space of X. It follows easily from classical properties of lifting maps to covering = 0 , there is a structure of topological group on X such that spaces ...

SCHEMES 01H8 Contents 1. Introduction 1 2. Locally ringed spaces

... locally generated by sections. Let i : Z → X be the associated closed subspace. A morphism f : Y → X factors through Z if and only if the map f ∗ I → f ∗ OX = OY is zero. If this is the case the morphism g : Y → Z such that f = i ◦ g is unique. Proof. Clearly if f factors as Y → Z → X then the map f ...

... locally generated by sections. Let i : Z → X be the associated closed subspace. A morphism f : Y → X factors through Z if and only if the map f ∗ I → f ∗ OX = OY is zero. If this is the case the morphism g : Y → Z such that f = i ◦ g is unique. Proof. Clearly if f factors as Y → Z → X then the map f ...

homotopy types of topological stacks

... homotopy type as a diagram of topological spaces. Furthermore, the transformation ϕ relates the given diagram of stacks with its weak homotopy type, thus allowing one to transport homotopical information back and forth between the diagram and its homotopy type. The above theorem has various applicat ...

... homotopy type as a diagram of topological spaces. Furthermore, the transformation ϕ relates the given diagram of stacks with its weak homotopy type, thus allowing one to transport homotopical information back and forth between the diagram and its homotopy type. The above theorem has various applicat ...

Andr´e-Quillen (co)Homology, Abelianization and Stabilization

... I Assume that subcategory Cab is also a model category with fibrations and weak equivalences as in C I Assume that the forgetful functor has a left adjoint (”abelianization”). Ab : C o ...

... I Assume that subcategory Cab is also a model category with fibrations and weak equivalences as in C I Assume that the forgetful functor has a left adjoint (”abelianization”). Ab : C o ...

First-Order Logical Duality Henrik Forssell

... {G : B → 2 G(¬b) = 1}. So XB = M (>) is compact with a basis of clopen sets. And if G, H : B → 2 are two distinct lattice morphisms, there exists b ∈ B so that G ∈ M (b) and H ∈ / M (b), so XB is Hausdorff. A compact Hausdorff space with a basis of clopen sets is a Stone space. In summary, then, any ...

... {G : B → 2 G(¬b) = 1}. So XB = M (>) is compact with a basis of clopen sets. And if G, H : B → 2 are two distinct lattice morphisms, there exists b ∈ B so that G ∈ M (b) and H ∈ / M (b), so XB is Hausdorff. A compact Hausdorff space with a basis of clopen sets is a Stone space. In summary, then, any ...

Compact topological semilattices

... a, b ∈ A imply c ∈ A. Definition (Nachbin, 1948): A pospace X said to be normally ordered (or monotone normal), if given two closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V . Notation: By I we denote the unit interval [0, 1 ...

... a, b ∈ A imply c ∈ A. Definition (Nachbin, 1948): A pospace X said to be normally ordered (or monotone normal), if given two closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V . Notation: By I we denote the unit interval [0, 1 ...

On Chains in H-Closed Topological Pospaces

... In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [3, 4, 7–10, 14, 17]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By a partial order on a set X we mean a reflexive, transitive and anti-s ...

... In this paper all topological spaces will be assumed to be Hausdorff. We shall follow the terminology of [3, 4, 7–10, 14, 17]. If A is a subset of a topological space X, then we denote the closure of the set A in X by cl X (A). By a partial order on a set X we mean a reflexive, transitive and anti-s ...

THE HOMOMORPHISMS OF TOPOLOGICAL GROUPOIDS 1

... It is easy to see that φ and ψ are topological groupoid homomorphisms over base Z such that ψ ◦ φ = id(k◦h)∗ (G) and φ ◦ ψ = idh∗ (k∗ (G)) . Therefore φ is an isomorphism of topological groupoids over the base Z . Proposition 3.5. Let (G, X) be a topological groupoid. Then the topological groupoids ...

... It is easy to see that φ and ψ are topological groupoid homomorphisms over base Z such that ψ ◦ φ = id(k◦h)∗ (G) and φ ◦ ψ = idh∗ (k∗ (G)) . Therefore φ is an isomorphism of topological groupoids over the base Z . Proposition 3.5. Let (G, X) be a topological groupoid. Then the topological groupoids ...

Chapter 1 Sheaf theory

... complex variables and holomorphic differential geometry. The theory is also essential to real analytic geometry. The theory of sheaves provides a framework for solving “local to global” problems of the sort that are normally solved using partitions of unity in the smooth case. In this chapter we pro ...

... complex variables and holomorphic differential geometry. The theory is also essential to real analytic geometry. The theory of sheaves provides a framework for solving “local to global” problems of the sort that are normally solved using partitions of unity in the smooth case. In this chapter we pro ...

Differential Algebraic Topology

... over a closed manifold of dimension > 0 is such a generalized manifold. There are several approaches in the literature in this direction but they are at a more advanced level. We hope it is useful to present an approach to ordinary homology which reﬂects the spirit of Poincaré’s original idea and i ...

... over a closed manifold of dimension > 0 is such a generalized manifold. There are several approaches in the literature in this direction but they are at a more advanced level. We hope it is useful to present an approach to ordinary homology which reﬂects the spirit of Poincaré’s original idea and i ...

Topology I - School of Mathematics

... just as well be included as one of the axioms). The same conclusion (as for a simple, closed, continuous curve) holds also for any “complete” curve in lR2, i.e. a simple, continuous, unboundedly extended, non-closed curve both of those ends go off to infinity, without nontrivial limit points in the ...

... just as well be included as one of the axioms). The same conclusion (as for a simple, closed, continuous curve) holds also for any “complete” curve in lR2, i.e. a simple, continuous, unboundedly extended, non-closed curve both of those ends go off to infinity, without nontrivial limit points in the ...

Localization of Ringed Spaces - Scientific Research Publishing

... Definition 1. A morphism f : A B of sheaves of rings on a space X is called a localization morphism3 iff there is a multiplicative subsheaf S A so that f is isomorphic to the localization A S 1 A of A at S .4 A morphism of ringed spaces f : X Y is called a localization morphism iff f # : f ...

... Definition 1. A morphism f : A B of sheaves of rings on a space X is called a localization morphism3 iff there is a multiplicative subsheaf S A so that f is isomorphic to the localization A S 1 A of A at S .4 A morphism of ringed spaces f : X Y is called a localization morphism iff f # : f ...

Localization of ringed spaces

... From the discussion above, we see that it is possible to describe X1 ×Y X2 , at least as a set, from the following data: (1) the ringed space fibered product X1 ×RS Y X2 (which carries the data of the rings OX1 ,x1 ⊗OY,y OX2 ,x2 as stalks of its structure sheaf) and (2) the subsets S(x1 , x2 ) ⊆ Spe ...

... From the discussion above, we see that it is possible to describe X1 ×Y X2 , at least as a set, from the following data: (1) the ringed space fibered product X1 ×RS Y X2 (which carries the data of the rings OX1 ,x1 ⊗OY,y OX2 ,x2 as stalks of its structure sheaf) and (2) the subsets S(x1 , x2 ) ⊆ Spe ...

Algebraic K-theory of rings from a topological viewpoint

... The abelian groups Ki (R) are homotopy groups of a space which is canonically associated with the general linear group GL(R) , i.e., with the group of invertible matrices, over the ring R . Therefore, several methods from homotopy theory produce interesting results in algebraic K -theory of rings. ...

... The abelian groups Ki (R) are homotopy groups of a space which is canonically associated with the general linear group GL(R) , i.e., with the group of invertible matrices, over the ring R . Therefore, several methods from homotopy theory produce interesting results in algebraic K -theory of rings. ...