By A. Chigogidze

ISBN-10: 0444822259

ISBN-13: 9780444822253

This can be a complete advent into the strategy of inverse spectra - a strong process effectively hired in quite a few branches of topology.The inspiration of an inverse series and its limits, first seemed within the recognized memoir via Alexandrov the place a unique case of inverse spectra - the so-called projective spectra - have been thought of. the concept that of an inverse spectrum in its current shape used to be first brought by way of Lefschetz. in the meantime, Freudental, had brought the concept of a morphism of inverse spectra. the principles of the full approach to inverse spectra have been laid down in those simple works.Subsequently, inverse spectra started to be extensively studied and utilized, not just within the quite a few significant branches of topology, but additionally in practical research and algebra. this isn't excellent contemplating the explicit nature of inverse spectra and the intense strength of the comparable techniques.Updated surveys (including proofs of numerous statements) of the Hilbert dice and Hilbert house manifold theories are integrated within the booklet. contemporary advancements of the Menger and N?beling manifold theories also are presented.This paintings considerably extends and updates the author's formerly released e-book and has been thoroughly rewritten for you to include new advancements within the box.

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**Example text**

Let f : X -* Y be a near-homeomorphism and Y be a Polish A N R-space. Then f is approximately soft. PROOF. Let A be a closed subset of a Polish space B and let maps ~: A ~ X and r B --~ Y such t h a t f ~ = r be given. Consider an arbitrary open cover U 9 coy(Y) and suppose t h a t )4) 9 coy(Y) is a star-refibement of/4. 22, there is an open refinement 1; 9 coy(Y) of )/V such t h a t the following condition is satisfied: 9 If one of the two arbitrarily given );-close maps into Y, defined on A, has an extension to B, then the second also has an extension to B.

FACTORIZING SPECTRA AND THE SPECTRAL THEOREM 27 Let X be a dense subspace of a Polish space X ' . Assume, additionally, t h a t dim X -- n. 2). q consisting of n-dimensional Polish spaces. 6, we see t h a t the spectra S x , x , and S contain isomorphic cofinal and w-closed subspectra. In particular, there is an n-dimensional G~-subspace of X ' containing X. T h e following s t a t e m e n t will be needed later in C h a p t e r 5. 11. Let f" P ~ Y be a map of a Polish space P into a metrizable compactum Y .

Let /3P denote the Stone-(Tech compactification of P. Obviously, dim fiR - dim P - n. 10, tiP = lim 8, where 8 - { Z a , P~a, A} is a factorizing w-spectrum consisting of n-dimensional metrizable compacta. Since P is Polish, we see t h a t P is the intersection of countably m a n y functionally open subsets o f / 3 P . Therefore there exists an index a0 E A such t h a t P Pao-l(Pao(P))" Consider now the s p e c t r u m S p = {Pa, q~,Ao}, where A0 = {a E A" a _> a0}, P~ = p~(P) and qa f~ = p~/P~ for each a , f l E Ao with a _

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