General Topology II: Compactness, Homologies of General SpacesA.V. Arhangel'skii Compactness is related to a number of fundamental concepts of mathemat ics. Particularly important are compact Hausdorff spaces or compacta. Com pactness appeared in mathematics for the first time as one of the main topo logical properties of an interval, a square, a sphere and any closed, bounded subset of a finite dimensional Euclidean space. Once it was realized that pre cisely this property was responsible for a series of fundamental facts related to those sets such as boundedness and uniform continuity of continuous func tions defined on them, compactness was given an abstract definition in the language of general topology reaching far beyond the class of metric spaces. This immensely extended the realm of application of this concept (including in particular, function spaces of quite general nature). The fact, that general topology provided an adequate language for a description of the concept of compactness and secured a natural medium for its harmonious development is a major credit to this area of mathematics. The final formulation of a general definition of compactness and the creation of the foundations of the theory of compact topological spaces are due to P.S. Aleksandrov and Urysohn (see Aleksandrov and Urysohn (1971)). |
Contents
Compactness | 3 |
Translated from the Russian | 4 |
2 Compactness and Products | 13 |
Copyright | |
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General Topology II: Compactness, Homologies of General Spaces A. V. Arhangel' skii Limited preview - 2012 |
General Topology II: Compactness, Homologies of General Spaces A.V. Arhangel'skii No preview available - 2011 |
Common terms and phrases
acyclic Aleksandrov Alexander-Spanier arbitrary Arhangel'skii and Ponomarev axiom of countability Bredon called cardinal invariants Čech cohomology chain complexes Chap closed subsets closed subspace closure cochains coefficients cohomology groups cohomology sequence cohomology theory coincides compact group compact Hausdorff extension compact supports compactum considered construction contains continuous mapping convergence Corollary countable base countable tightness defined denote dense dimension Dokl duality Dugundji compact dyadic embedded Engelking 1977 English translation exact sequence example exists finite flabby following result formulas Fréchet-Urysohn functor group G Hausdorff space homology and cohomology homomorphism infinite intersection inverse isomorphism locally compact spaces manifolds Math metrizable Nauk neighborhood obtained open cover open subsets pair paracompact space particular polyhedra Ponomarev 1974 presheaf Proposition pseudocompact space resolution satisfying Sect sections sheaf G sheaves simplices singular spectral sequence Stone-Čech compactification T₁-space Theorem Tikhonov cube Tikhonov space topological group topological spaces zero-dimensional