General Topology I, Volumes 1-3
A. V. Arkhangelʹskiĭ, Lev Semenovich Pontri͡a͡gin
Springer-Verlag, 1990 - Mathematics - 202 pages
This is the first of the encyclopaedia volumes devoted to general topology. It has two parts. The first outlines the basic concepts and constructions of general topology, including several topics which have not previously been covered in English language texts. The second part presents a survey of dimension theory, from the very beginnings to the most important recent developments. The principal ideas and methods are treated in detail, and the main results are provided with sketches of proofs. The authors have suceeded admirably in the difficult task of writing a book which will not only be accessible to the general scientist and the undergraduate, but will also appeal to the professional mathematician. The authors' efforts to detail the relationship between more specialized topics and the central themes of topology give the book a broad scholarly appeal which far transcends narrow disciplinary lines.
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Arkhangelskii V V Fedorchuk
2 Some Important Classes of Topological Spaces
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Aleksandrov axiom of countability called Cantor manifold Cantor perfect set closed map closed sets closed subsets co-map cohomological dimension collection compactification completely regular constructed contains continuous map converges Corollary countable base countable dimensional decomposition space defined Definition denote dimension dim dimension Ind dimension less dimension theory dimensional compacta dimG dimR dimz equal essential map Euclidean space everywhere dense Example exists Fedorchuk function Hausdorff space hereditarily normal space homeomorphic inductive dimensions inequality intersection inverse system Lemma locally compact space locally finite map f metrizable compactum multiplicity less n-dimensional neighbourhood nonempty obtain open cover open map open sets open subsets paracompact space Pasynkov plane point x e polyhedron preimage proof Proposition proved quotient map refinement satisfies segment sequence sequential spectrum subspace Theorem 16 Tikhonov topological space topology 9 transfinite dimension triangulation uniform space union Uryson vertices weakly infinite dimensional zero-dimensional map