Category Of Topological Spaces Research Articles

A cartesian closed category for topology

By replacing sequences by sets in definitions of Fréchet and Urysohn, one obtains the definition of an L ∗ space. The category of L ∗ spaces properly contains the category of topological spaces, it is cartesian closed, and it has other properties which make it convenient for topology. The category of L ∗ spaces is properly smaller than the category of convergence spaces (Limesräame) of Cook and Fischer, but properly larger than the category of epitopological spaces of Antoine.

The non-simplicity of certain categories of topological spaces

The non-simplicity of certain categories of topological spaces

Remark on a theorem of E.H. Brown

The proliferation of classifying spaces in recent years owes much to the theorem of Edgar H. Brown, Jr on the representability of homotopy functors. Since the theorem only gives a representation for functors defined on the category of spaces having the homotopy type of a CW complex, there is some interest in finding conditions under which the domain category may be enlarged. It appears that a version of the theorem holds for any small full subcategory of Htp, the category of topological spaces and homotopy classes of continuous maps, but that the resulting classifying space is generally intractable.

The Homology of Uniform Spaces

In the past thirty years, algebraic topologists have developed a great body of knowledge concerning the category of topological spaces. By contrast, corresponding problems in the category of uniform spaces have been barely touched. Lubkin [8] studied the notion of a covering space in the category of generalized uniform spaces, and suggested that much of algebraic topology could be profitably studied in this category. Deming [2] discussed the fundamental group of a generalized uniform space, and related it to the first Čech homology group. A slightly different version of Cech cohomology was defined by Kuzminov and Svedov in [7] and related to the dimension theory of uniform spaces.

On topological quotient maps preserved by pullbacks or products

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?

Prime ideal structure in commutative rings

0. Introduction. Let ' be the category of commutative rings with unit, and regard Spec (as in [1]) as a contravariant functor from ' to g$7 the category of topological spaces and continuous maps. The spaces of the form Spec A, A a ring (ring always means object of '), are well known to have many special properties. It is worthwhile to discover whether these well-known properties of the spaces characterize them, since we then will know the limitations of the topological approach. As a by-product of this study, remarkable facts about the structure of the prime ideals in a commutative ring come to light. E.g. given any ring A there is a ring whose prime ideals have precisely the reverse order of the primes of A. In the same vein, on the topological level there is a complete duality between localization and taking residue class rings (see ?8, Proposition 8). We call a space spectral if it is To and quasi-compact, the quasi-compact open subsets are closed under finite intersection and form an open basis, and every nonempty irreducible closed subset has a generic point. Spec A, A a ring, is well known to be spectral, and in fact these properties do characterize the spaces in the image of Spec. However, there are much more enlightening characterizations. For example, the last property on the list is a very special case of a much more general property of spectral spaces-best described as the compactness of a new topology on the spectral space which is derived from the original topology. Again, the spectral spaces are precisely the projective limits of finite To spaces. Proving that these properties characterize the image of Spec involves constructing a large number of rings. This construction (??3-7) is very intricate, but is practically choice-frep and turns out to be fairly functorial. Moreover, we can fix any field k and get the constructed rings to be k-algebras. To be more precise about the functorial aspects, we define a continuous map of spectral spaces to be spectral if inverse images of quasi-compact open sets are quasi-compact. Let Y be the subcategory of $-consisting of spectral spaces and spectral maps. Then it is well known that every space and map in the image of Spec is in Y and it will turn out that up to isomorphism, every space and map in Y is in the image of Spec. In this sense, we may say that Y is precisely the image of Spec. We say that Spec is invertible on a subcategory I of Y if there is a (contravariant) functor F: q -W' whose composition with Spec is isomorphic with the inclusion

The Coreflective Subcategory of Sequential Spaces

Certain theorems of recent interest [1, 2] concerning sequential spaces may be deduced from the fact that the category of sequential spaces, is a coreflective subcategory of the category of topological spaces, J. A space is said to be sequential if it has the finest topology that permits the convergence of its convergent sequences.

Microbundles and bundles

In this part I we work in two categories namely the category of topological spaces and continuous maps, and the category of piecewise linear (PL) spaces and PL-maps. A PL-space is a topological space with a complete class of locally finite triangulations, any two of which have a common subdivision. A PL-mapf : X-~ Yis a continuous map between PL-spaces X and Y, which for some triangulation of X and Y maps every simplex of X linearly into a simplex of Y. We consider fibrebundles in the sense of STEENROD with a fixed crosssection, often called the zeroseetion,

Dihomology III. A Generalization of the Poincaré Duality for Manifolds

which is a topological invariant of the space X, but not an invariant of homotopy type. The E term is the Cech cohomology of X with coefficients in the sheaf £ of local singular homology of X, and the ETM term is related to the ordinary global singular homology of X. The sequence therefore relates the local and global structures of the space. If X is a closed orientable w-manifold, then the local homology sheaf reduces to the simple sheaf of integers, and the spectral sequence collapses to the familiar isomorphism of Poincare, H^Hn_Q. If X is a polyhedron, there is a simple way of defining E, and its dual $, using a triangulation. In order to obtain the spectral sequences as quickly as possible we give this simplicial method in §2, and summarize the properties of the sequences in Theorem 1. In §3 we generalize E,£ to arbitrary topological spaces, using a combination of singular homology and Cech cohomology, and verify that the simplicial sequences are in fact topological invariants. The functorial qualities of E,$ are also discussed. Since both E and $ involve both homology and cohomology, they are not functors on the category of topological spaces but we prove in Theorem 2 that if we generalize them further they are functors on a category of maps, the sequences of a space being those associated with the identity map. The last section is concerned with the geometrical interpretation of E and $, and, in particular, of the filtrations induced on the homology and cohomology groups. It appears that the filtration of a homology or cohomology class has something to do with the dimension of that part of the space in which it is 'situated', and we prove two theorems which are feelers in this direction. Theorem 3 connects the homology filtration with cap products, while Theorem 4 concerns the filtration of a cohomology

GFB-NBS strain tester for rubber and rubber-like compounds

Computational effects can be modelled by observationally-induced algebras, which are algebras whose structure is completely determined by a chosen computational prototype. We show that the category of continuous maps between topological spaces supports a free observationally-induced algebra construction for arbitrary pre-chosen prototype, and give a characterisation of the free algebras as subalgebras of certain powers of these prototypes.Moreover, we present observationally-induced lower and upper powerspace constructions in the category of topological spaces. Our lower powerspace construction is for all topological spaces given by its non-empty closed subsets equipped with the lower Vietoris topology. Dually, our upper powerdomain construction is for a wide class of topological spaces given by the space of proper open filters of its topology equipped with the upper Vietoris topology. Thus, both constructions generalise the classical construction on continuous dcpos, and unify abstract and concrete characterisations of powerdomains on a broader scale.Finally, we show that with a small adjustment of the definitions, observationally-induced algebras form the smallest full reflective subcategory of the category of algebras for the corresponding signature, which contains the computational prototype and is complete and closed under isomorphism.