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# Topic: Tychonoff

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 math lessons - Andrey Nikolayevich Tychonoff Andrey Nikolayevich Tychonoff (Андрей Николаевич Тихонов: October 30, 1906–1993) was a Russian mathematician. Tychonoff originally published in German, whence the transliteration. Tychonoff worked in a number of different fields in mathematics. www.mathdaily.com /lessons/Andrey_Nikolayevich_Tychonoff   (206 words)

 Tychonoff's theorem   (Site not responding. Last check: 2007-10-19) Tychonoff's theorem in topology states that the product of any collection of compact topological spaces is compact. Tychonoff's theorem is complex, and its proof is often approached in parts, proving helpful lemmas first. To prove that Tychonoff's theorem implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty. bopedia.com /en/wikipedia/t/ty/tychonoff_s_theorem.html   (754 words)

 Kids.Net.Au - Encyclopedia > Tychonoff space   (Site not responding. Last check: 2007-10-19) In topology and related brances of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. Tychonoff spaces are named after Andrey Tychonoff[?], whose Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov". Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space. www.kids.net.au /encyclopedia-wiki/ty/Tychonoff_space   (426 words)

 Tychonoff_space - The Wordbook Encyclopedia Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (???????) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov". The Niemytzki plane is an example of a Tychonoff space which is not normal. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K and an injective continuous map j from X to K such that the inverse of j is also continuous. www.thewordbook.com /Tychonoff_space   (677 words)

 Tychonoff's theorem - Wikipedia, the free encyclopedia This theorem of Tychonoff has many applications in differential and algebraic topology and in functional analysis, e.g., for the Stone-Čech compactification or in the proof of the Theorem of Banach-Alaoğlu. To actually prove Tychonoff's theorem, we use the definition of compactness based on the FIP, by taking an FIP collection A of sets, and showing that the intersection over closures of elements of A is nonempty. To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty. en.wikipedia.org /wiki/Tychonoff's_theorem   (1217 words)

 PlanetMath: proof of Tychonoff's theorem in finite case (The finite case of Tychonoff's Theorem is of course a subset of the infinite case, but the proof is substantially easier, so that is why it is presented here.) "proof of Tychonoff's theorem in finite case" is owned by stevecheng. This is version 1 of proof of Tychonoff's theorem in finite case, born on 2005-08-03. planetmath.org /encyclopedia/ProofOfTychonoffsTheoremInFiniteCase.html   (185 words)

 Tychonoff space - ExampleProblems.com Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov". X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K and an injective continuous map j from X to K such that the inverse of j is also continuous. www.exampleproblems.com /wiki/index.php/Tychonoff_space   (583 words)

 [No title]   (Site not responding. Last check: 2007-10-19) In fact, so is Tychonoff's Theorem, as it can be used to prove the Axiom of Choice; the result is due to (I believe) Kelley. Note that Tychonoff's theorem for T1 spaces (points are closed) is equivalent to AC. For the more common application involving Hausdorff spaces, the theorem is equivalent to "the Prime Ideal Theorem", which says, in one form, that every non-trivial filter in a Boolean algebra can be extended to an ultrafilter. www.math.niu.edu /~rusin/known-math/95/tychonoff   (488 words)

 Summary   (Site not responding. Last check: 2007-10-19) Tychonoff's theorem in the framework of formal topologies, by S. Negri and S. Valentini, The Journal of Symbolic Logic, 62,4, pp. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space. Tychonoff's theorem is then proved both for the binary and arbitrary product of topological spaces. www.helsinki.fi /~negri/summary-tyc.html   (398 words)

 Tychonoff Theorem and AC in locales by Martín Escardó The localic proofs of Tychonoff are different from the topological ones (i.e., they are not special cases where a choice function is supplied). In other words, Tychonoff shows that a product of compact locales is compact, but doesn't say anything about the spatiality of the product. By ignoring the question of whether the frames that one has at hand happen to be subframes of powersets, the axiom of choice is avoided in several theorems in locale theory. at.yorku.ca /t/o/p/d/55.htm   (1313 words)

 Tychonoff space - Wikipedia, the free encyclopedia X is a completely regular space if and only if, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) is 0 and f(y) is 1 for every y in F. A topological space is Tychonoff if and only if it's both completely regular and T The Niemytzki plane is an example of a Tychonoff space which is not normal. en.wikipedia.org /wiki/Tychonoff_space   (629 words)

 Atlas: Tychonoff Convergence Spaces by Mehmet Baran   (Site not responding. Last check: 2007-10-19) In particular, there are non-constant continuous functions on every Tychonoff space into the real numbers with the usual topology. Tychonoff spaces are, moreover, the most general topological spaces that can be guaranteed to have this property. In earlier papers, various generalizations of Tychonoff objects for an arbitrary topological category were defined. atlas-conferences.com /c/a/g/x/18.htm   (231 words)

 topologyHW_II The main lemma that implies the Tychonoff theorem is the following: A topological space is compact if every subbasic open cover has a finite subcover. [HINT: It is a fact that the Tychonoff theorem does not generalize to Lindelöf spaces. I.E. the product of Lindelöf spaces (with the Tychonoff topology) may not be Lindelöf (see for example, Kelley, "General Topology" page 59, Problem L where there is an example of a Lindelöf space, math.hunter.cuny.edu /~mbenders/topologyHW_II   (224 words)

 Topology MAT 530 Some set theoretic preparation for the proof of the Tychonoff theorem is here [pdf, ps]. Although the Tychonoff topology is very coarse, it turns out to be just the right topology for many purposes. The Tychonoff theorem states that the product of any number of compact spaces is compact in the Tychonoff topology. www.math.sunysb.edu /~timorin/mat530.html   (2896 words)

 On Tychonoff-type hypertopologies by Georgi Dimov, Franco Obersnel and Gino Tironi   (Site not responding. Last check: 2007-10-19) In 1975, M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. Tychonoff topology, Tychonoff-type topology, T-space, commutative space, $\mathcal{O}$-commutative space, $\mathcal{M}$-cover, $\mathcal{M}$-closed family, $P_\infty$-space. www.emis.de /proceedings/TopoSym2001/06.htm   (180 words)

 [No title] It is not hard to see that that a regular space in which every point has a Tychonoff neighborhood is already a Tychonoff space. When X is Tychonoff, the map i is a homeomorphism onto its image, and we identify X with i(X). Proof The space E is locally Tychonoff, so it will suffice to show that E is regular. hopf.math.purdue.edu /Feldman-Wilce/fibdegen.txt   (5513 words)

 LatexWiki Website - EasyProofOfTychonoff   (Site not responding. Last check: 2007-10-19) Tychonoff's theorem says that the product of compact topological spaces is compact. Lemma (Alexander): If X is a topological space and S is a subbasis of X, then X is compact if and only if every S-cover of X has a finite subcover. Plone makes heavy use of CSS, which means it is accessible to any internet browser, but the design needs a standards-compliant browser to look like we intended it. latexwiki.rootnode.com /wiki/EasyProofOfTychonoff   (434 words)

 On Tychonoff-type hypertopologies by Georgi Dimov, Franco Obersnel and Gino Tironi   (Site not responding. Last check: 2007-10-19) In 1975, M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. Tychonoff topology, Tychonoff-type topology, T-space, commutative space, $\mathcal{O}$-commutative space, $\mathcal{M}$-cover, $\mathcal{M}$-closed family, $P_\infty$-space. www.univie.ac.at /EMIS/proceedings/TopoSym2001/06.htm   (180 words)

 Oleg Pavlov   (Site not responding. Last check: 2007-10-19) In particular, for any Tychonoff non-pseudocompact space $X$ there is a $\mu$ such that $X^\mu$ can be condensed onto a normal ($\sigma$-compact) space if and only if there is no measurable cardinal. For any Tychonoff space $X$ and any cardinal $\nu$ there is a Tychonoff space $M$ which preserves many properties of $X$ and such that any one-to-one continuous image of $M^\mu$, $\mu \leq \nu$, contains a closed copy of $X^\mu$. For any infinite compact space $K$ there is a normal space $X$ such that $X\times K$ cannot be mapped one-to-one onto a normal space. www.maths.tcd.ie /EMIS/journals/CMUC/cmuc9902/abs/pavlov.htm   (117 words)

 Tychonoff's Theorem (ResearchIndex)   (Site not responding. Last check: 2007-10-19) Even the definition of the product topology is an interesting formalisation problem in type theory. Known proofs of the localic Tychonoff's theorem, such as [2] or [1] a priori uses a decidability hypothesis on the index... 5 An Intuitionistic Proof of Tychonoff's Theorem (context) - Coquand - 1991 DBLP citeseer.ist.psu.edu /coquand97tychonoffs.html   (242 words)

 Citations: Differentialgleichungen der mathematischen Physik - Tychonoff, Samarski (ResearchIndex)   (Site not responding. Last check: 2007-10-19) Citations: Differentialgleichungen der mathematischen Physik - Tychonoff, Samarski (ResearchIndex) A.N. Tychonoff and A.A. Samarski, Differentialgleichungen der mathematischen Physik, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959. The string material is characterized by its density ae and its Young s modulus E. is the tension applied to the string and d1 is a frequency.... citeseer.ist.psu.edu /context/683517/0   (273 words)

 Atlas: Constructing Tychonoff $G$-spaces which are not $G$-Tychonoff by Tzvi Scarr   (Site not responding. Last check: 2007-10-19) If X is Tychonoff (respectively, normal, compact, etc.), then (or just X, for short) is called a Tychonoff (respectively, normal, compact, etc.) G-space. Our modification converts a normal G-space which is not G-normal in some sense into a Tychonoff G-space which is not G-Tychonoff. The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caai-98. atlas-conferences.com /c/a/a/i/98.htm   (370 words)

 Product topology - Wikipedia, the free encyclopedia The product topology is sometimes called the Tychonoff topology. An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. This is easy to show for finite products, while the general statement is equivalent to the axiom of choice. en.wikipedia.org /wiki/Product_space   (835 words)

 @CAT 2004-2005 The spaces X whose reflection X --> beta X is an embedding are precisely the Tychonoff spaces (= T_1 and completely regular). In case of topological groups, however, the situation is more complex, because every T_0 group is Tychonoff. The category Grp(CompHaus) of compact Hausdorff groups is reflective in Grp(Top), but the groups G whose reflection rho: G --> bG is an embedding is a much narrower class that those of the Tychonoff groups. www.mscs.dal.ca /~pare/Sem04-05.html   (1980 words)

 Mathematics 705: Analysis III   (Site not responding. Last check: 2007-10-19) Ascoli A proof of the Ascoli Theorem the Tychonoff product theorem. Tychonoff Some standard applications of the Tychonoff product theorem. Vitali A proof of the Vitali covering theorem for measures on metric spaces that satisfy a doubling condition. www.math.sc.edu /~howard/Classes/705   (56 words)

 [No title] If a space X has a (Hausdroff) compactification, then X must be a Tychonoff space. A compactification of a space X is a pair (K,e) where K is a compact space and e is an embedding of X as a dense subset of K. This is clear since it is well know that Compact spaces are themselves Tychonoff and subspaces of Tychonoff spaces are also Tychonoff. Because X is Tychonoff C^*(X) separates points from closed sets, so by the embedding theorem [16:51] br.endernet.org /~loner/topology/stonecech.txt   (2630 words)

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