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1. Lectures on Ordinary Differential Equations (Dover Phoenix Editions) (Dover Phoneix Editions) by Witold Hurewicz | |
Hardcover: 144
Pages
(2002-05-01)
list price: US$27.50 Isbn: 0486495108 Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description |
2. Collected Works of Witold Hurewicz (Collected Works, Vol.4) by Krystyna Kuperberg | |
Hardcover: 598
Pages
(1995-08-01)
list price: US$226.00 -- used & new: US$205.00 (price subject to change: see help) Asin: 0821800116 Canada | United Kingdom | Germany | France | Japan | |
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3. Dimension Theory (Princeton Mathematical Ser.; Vol 4) by Witold Hurewicz, Henry Wallman | |
Hardcover: 176
Pages
(1996-12)
list price: US$49.50 Isbn: 0691079471 Average Customer Review: Canada | United Kingdom | Germany | France | Japan | |
Customer Reviews (4)
A great book: Dimension Theory by Hurewicz and Wallman
The standard treatise on classical dimension theory
Still an excellent book In chapter 2, the authors concern themselves with spaces having dimension 0. They first define dimension 0 at a point, which means that every point has arbitrarily small neighborhoods with empty boundaries. A 0-dimensional space is thus 0-dimensional at every one of its points. Several examples are given (which the reader is to prove), such as the rational numbers and the Cantor set. It is shown, as expected intuitively, that a 0-dimensional space is totally disconnected. The authors also show that a space which is the countable sum of 0-dimensional closed subsets is 0-dimensional. The closed assumption is necessary here, as consideration of the rational and irrational subsets of the real line will bring out. Chapter 3 considers spaces of dimension n, the notion of dimension n being defined inductively. Their definition of course allows the existence of spaces of infinite dimension, and the authors are quick to point out that dimension, although a topological invariant, is not an invariant under continuous transformations. The famous Peano dimension-raising function is given as an example. The authors prove an equivalent definition of dimension, by showing that a space has dimension less than or equal to n if every point in the space can be separated by a closed set of dimension less than or equal to n-1 from any closed set not containing the point. The `sum theorem' for dimension n is proven, which says that a space which is the countable union of closed sets of dimension less than or equal to n also has dimension less than or equal to n. A successful theory of dimension would have to show that ordinary Euclidean n-space has dimension n, in terms of the inductive definition of dimension given. The authors show this in Chapter 4, with the proof boiling down to showing that the dimension of Euclidean n-space is greater than or equal to n. (The reverse inequality follows from chapter 3).The proofofthis involves showing that the mappings of the n-sphere to itself which have different degree cannot be homotopic. The authors give an elementary proof of this fact. This chapter also introduces the study of infinite-dimensional spaces, and as expected, Hilbert spaces play a role here. The Lebesgue covering theorem, which was also proved in chapter 4, is used in chapter 5 to formulate a covering definition of dimension. The author also proves in this chapter that every separable metric space of dimension less than or equal to n can be topologically imbedded in Euclidean space of dimension 2n + 1. The author quotes, but unfortunately does not prove, the counterexample due to Antonio Flores, showing that the number 2n + 1 is the best possible. These considerations motivate the concept of a universal n-dimensional space, into which every space of dimension less than or equal to n can be topologically imbedded. The author also proves a result of Alexandroff on the approximation of compact spaces by polytopes, and a consequent definition of dimension in terms of polytopes. Chapter 6 has the flair of differential topology, wherein the author discusses mappings into spheres. This brings up of course the notion of a homotopy, and the author uses homotopy to discuss the nature of essential mappings into the n-sphere. The author motivates the idea of an essential mapping quite nicely, viewing them as mappings that cover a point so well that the point remains covered under small perturbations of the mapping. This chapter also introduces extensions of mappings and proves Tietze's extension theorem. This allows a characterization of dimension in terms ofthe extensions of mappings into spheres, namely that a space has dimension less than or equal to n if and only if for every closed set and mapping from this closed set into the n-sphere, there is an extension of this mapping to the whole space. In chapter 7 the author relates dimension theory to measure theory, and proves that a space has dimension less than or equal to n if and only if it is homeomorphic to a subset of the (2n+1)-dimensional cube whose (n+1)-dimensional measure is zero. As a sign of the book's age, only a short paragraph is devoted to the concept of Hausdorff dimension. Hausdorff dimension is of enormous importance today due to the interest in fractal geometry. Chapter 8 is the longest of the book, and is a study of dimension from the standpoint of algebraic topology. The treatment is relatively self-contained, which is why the chapter is so large, and the author treats both homology and cohomology.The author proves that a compact space has dimension less than or equal to n if and only if given any closed subset, the zero element of the n-th homology group of this subset is a boundary in the space. A similar (dual) result is proven using cohomology.
Complete survey of dimension theory up to 1940. Please read my other reviews in my member page (just click onmy name above). ... Read more |
4. Lectures on Ordinary Differential Equations by Witold Hurewicz | |
Hardcover:
Pages
(1958-01-01)
Asin: B001M0CC6M Canada | United Kingdom | Germany | France | Japan | |
5. Lectures on Ordinary Differential Equati by Witold Hurewicz | |
Hardcover:
Pages
(1980)
Asin: B000N7BQQW Canada | United Kingdom | Germany | France | Japan | |
6. Dimension Theory Rev Edition by Witold Hurewicz | |
Hardcover:
Pages
(1948-01-01)
Asin: B001TGNP1K Canada | United Kingdom | Germany | France | Japan | |
7. Witold Hurewicz: An entry from Gale's <i>Science and Its Times</i> | |
Digital: 1
Pages
(2000)
list price: US$0.98 -- used & new: US$0.98 (price subject to change: see help) Asin: B0027UWWE8 Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description |
8. Witold Hurewicz | |
Paperback: 82
Pages
(2010-08-05)
list price: US$45.00 -- used & new: US$41.38 (price subject to change: see help) Asin: 613112003X Canada | United Kingdom | Germany | France | Japan | |
Editorial Review Product Description |
9. Reprint: "Witold Hurewicz, in Memoriam" by Solomon Lefschetz | |
Paperback:
Pages
(1957-01-01)
Asin: B0038U0GVC Canada | United Kingdom | Germany | France | Japan | |
10. Lectures on Ordinary Differential Equations by Witold Hurewicz | |
Hardcover: 122
Pages
(1958)
Asin: B0000CK1W7 Canada | United Kingdom | Germany | France | Japan | |
11. Lectures on Ordinary Differential Equations by witold hurewicz | |
Paperback:
Pages
(1958-01-01)
Asin: B002I9VQBS Canada | United Kingdom | Germany | France | Japan | |
12. Ordinary differential equations in the real domain with emphasis on geometric methods by Witold Hurewicz | |
Unknown Binding:
Pages
(1956)
Asin: B0007FID1Q Canada | United Kingdom | Germany | France | Japan | |
13. Ordinary differential equations by Witold Hurewicz | |
Unknown Binding: 122
Pages
(1958)
Asin: B0007GWW1W Canada | United Kingdom | Germany | France | Japan | |
14. Ordinary Differential Equations in the Real Domain with Emphasis on Geometric Methods by Witold Hurewicz | |
Hardcover:
Pages
(1943)
Asin: B001JZ0FV4 Canada | United Kingdom | Germany | France | Japan | |
15. Dimension Theory by Witold & Henry Wallman Hurewicz | |
Hardcover:
Pages
(1974)
Asin: B001RHVPJK Canada | United Kingdom | Germany | France | Japan | |
16. Ordinary Differential Equations in the R by Witold Hurewicz | |
Hardcover:
Pages
(1943)
Asin: B000Q9YRJU Canada | United Kingdom | Germany | France | Japan | |
17. dimension Theory by Witold and Henry Wallman Hurewicz | |
Hardcover:
Pages
(1968)
Asin: B000HKIICA Canada | United Kingdom | Germany | France | Japan | |
18. Dimension Theory by Witold Hurewicz | |
Paperback:
Pages
(1941)
Asin: B001IJRR44 Canada | United Kingdom | Germany | France | Japan | |
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