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# Topic: Tychonoff space

###### In the News (Tue 25 Jun 19)

 Tychonoff space - Wikipedia, the free encyclopedia 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. Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space. 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. en.wikipedia.org /wiki/Tychonoff_space   (624 words)

 Locally compact space - Wikipedia, the free encyclopedia Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article Compact space. Thus locally compact spaces are as useful in p-adic analysis as in classical analysis. As mentioned in the previous section, any compact Hausdorff space is also locally compact, and any locally compact Hausdorff space is in fact a Tychonoff space. en.wikipedia.org /wiki/Locally_compact_space   (1352 words)

 Encyclopedia: Metric space   (Site not responding. Last check: 2007-10-21) A metrizable space is a topological space that is homeomorphic to a metric space. If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X. So to decide in what sense two metrics spaces are equivalent we have to discuss continuous functions between them (morphisms preserving the topology of the metric spaces). www.nationmaster.com /encyclopedia/metric-space   (4193 words)

 Compact space In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R A metric space is compact if and only if every sequence in the space has a subsequence with limit in the space. A topological space is compact if and only if every net on the space has a subnet which has a limit in the space. www.ebroadcast.com.au /lookup/encyclopedia/co/Compactness.html   (802 words)

 Normal space - Wikipedia X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E and a neighbourhood V of F that are also disjoint. Sierpinski space[?] is an example of a normal space that isn't regular. A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence[?]. wikipedia.findthelinks.com /no/Normal_Hausdorff_space.html   (897 words)

 Compact space   (Site not responding. Last check: 2007-10-21) Suppose X is a Hausdorff space, and we have a point x in X and a finite subset A of X not containing x. A topological space X is called compact iff all its open covers have a finite subcover. A metric space (or more generally any first countable space) is compact if and only if every sequence in the space has a convergent subsequence. www.seepage.org /wiki/index.php?title=Compact_space   (1344 words)

 Topology glossary A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are functionally separated. A partition of unity of a space X is a set of continuous functions from X to [0,1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods. www.teachersparadise.com /ency/en/wikipedia/t/to/topology_glossary.html   (2497 words)

 Completely Hausdorff space - One Language   (Site not responding. Last check: 2007-10-21) In topology and related fields of mathematics, a completely Hausdorff space is a type of Hausdorff space satisfying a slightly stronger separation axiom. Most spaces studied in mathematics satisfy strictly weaker conditions (such as being Hausdorff), strictly stronger conditions (such as being Tychonoff), or unrelated conditions (such as being regular). An example of a space which satisfies precisely this condition is the cocountable extension topology, which is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. www.onelang.com /encyclopedia/index.php/Completely_Hausdorff_space   (397 words)

 Encyclopedia article on Topological space [EncycloZine]   (Site not responding. Last check: 2007-10-21) Any metric space turns into a topological space if one defines the open sets to be generated by the set of all open balls. A linear graph is a topological space that generalises many of the geometric aspects of graphs with vertices and edges. Topological spaces can be broadly classified according to their degree of connectedness, their size, their degree of compactness and the degree of separation of their points and subsets. encyclozine.com /Topological_space   (2350 words)

 Some Questions and References on Relative Topological Properties, Part 1 Let Y be a (dense) subspace of a Tychonoff space X such that X is normal and countably paracompact (1-countably paracompact) on Y. Is then true that X x I is normal on Y x I? (Where I is the closed interval [0, 1]). Let Y be a (dense) subspace of a Tychonoff space X such that X x B is normal on Y x B, for each compact Hausdorff space B. Is then Y paracompact (1-paracompact) in X? Problem 17. Let Y be a (dense) subspace of a Tychonoff space X such that Y is internally normal in X and bounded in X. Is then Y countably compact in X? Problem 18. at.yorku.ca /i/a/a/i/04.htm   (728 words)

 PlanetMath: Tychonoff space (and therefore Hausdorff) is called a Tychonoff space, or a This is version 4 of Tychonoff space, born on 2002-01-22, modified 2006-04-03. Object id is 1534, canonical name is Tychonoff. planetmath.org /encyclopedia/Tychonoff.html   (58 words)

 PlanetMath: T2 space space is also known as a Hausdorff space. Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces. This is version 15 of T2 space, born on 2002-02-08, modified 2005-05-10. planetmath.org /encyclopedia/Hausdorff.html   (303 words)

 Shinji Kawaguchi, Ryoken Sokei   (Site not responding. Last check: 2007-10-21) Abstract:Paracompactness ($=2$-paracompactness) and normality of a subspace $Y$ in a space $X$ defined by Arhangel'skii and Genedi [4] are fundamental in the study of relative topological properties ([2], [3]). In fact, Bella and Yaschenko [8] characterized Tychonoff spaces which are normal in every larger Tychonoff space, and this result is essentially implied by their previous result in [8] on a corresponding case of weak $C$-embeddings. Finally, we construct a Tychonoff space $X$ and a subspace $Y$ such that $Y$ is $1$-paracompact in $X$ but not $1$-subparacompact in $X$. www.emis.de /journals/CMUC/cmuc0503/abs/kawasoke.htm   (257 words)

 Stone-Weierstrass theorem The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated. Further, there is a generalization of the Stone-Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space space is approximated uniformly on compact sets by algebras of the type appearing in the Stone-Weierstrass theorem and described below. As a consequence of the Weierstrass approximation theorem, one can show that the space C[a,b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. www.knowledgefun.com /book/s/st/stone_weierstrass_theorem.html   (907 words)

 Oleg Pavlov   (Site not responding. Last check: 2007-10-21) 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 space: Facts and details from Encyclopedia Topic   (Site not responding. Last check: 2007-10-21) Tychonoff spaces and completely regular spaces are particularly nice kinds of topological space ((mathematics) any set of points that satisfy a set of postulates of some kind) Almost every topological space studied in mathematical analysis[follow this hyperlink for a summary of this topic] is Tychonoff, Exception Handler: No article summary found. Tychonoff spaces are precisely those topological space ((mathematics) any set of points that satisfy a set of postulates of some kind) www.absoluteastronomy.com /ref/tychonoff_space   (2080 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). When H is a p-group, it is a consequence of Miller's version of the Sullivan conjecture that the homotopy fixed point space of (H)(n), that is, Hom H (EH; (H)(n)) is homotopy equivalent to the fixed point space of (H)(n)[[1], Theorem A]. www.math.purdue.edu /research/atopology/Feldman-Wilce/fibdegen.txt   (5513 words)

 HJM, Vol. 29, No. 2, 2003 A submanifold of a pseudo-Euclidean space is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. The Space of Limits of Continua in the Fell Topology, pp. In this paper we shall extend this theory to a suitable Boehmian space and identify a subspace of this Boehmian space on which the Hilbert transform becomes a one-to-one continuous linear map. www.math.uh.edu /~hjm/Vol29-2.html   (1685 words)

 ipedia.com: Axiom of choice Article   (Site not responding. Last check: 2007-10-21) The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem. Tychonoff's theorem stating that every product of compact topological spaces is compact. A uniform space is compact if and only if it is complete and totally bounded. www.ipedia.com /axiom_of_choice.html   (1185 words)

 A. Tamariz-Mascar\'ua, H. Villegas-Rodr\'{\i }guez   (Site not responding. Last check: 2007-10-21) A space $X$ is called {almost-$\omega$-resolvable} provided that $X$ is the union of a countable increasing family of subsets each of them with an empty interior. We prove that every almost-$\omega$-resolvable space is $C_\square$-discrete, and that these classes coincide in the realm of completely regular spaces. Finally, we prove that it is consistent with $ZFC$ that every dense-in-itself space is almost-$\omega$-resolvable, and that the existence of a measurable cardinal is equiconsistent with the existence of a Tychonoff space without isolated points which is not almost-$\omega$-resolvable. www.maths.tcd.ie /EMIS/journals/CMUC/cmuc0204/abs/Tamavil.htm   (184 words)

 On the posets of all the Hausdorff and all the Tychonoff compactifications of mappings In this paper, a space means a topological space and a mapping is a continuous function. It is well-known that the poset of all Hausdorff compactifications of a Tychonoff space is a complete upper semilattice. Let us note that we consider both Hausdorff and Tychonoff compactifications of mappings because, unlike the case of spaces, there are compact Hausdorff mappings which are not Tychonoff (see [HI]and [C]). giorgionordo.equal.it /ricerca/lavori/bn.htm   (1553 words)

 Emad Abu Osba, Melvin Henriksen   (Site not responding. Last check: 2007-10-21) A point $p$ of a (Tychonoff) space $X$ is called a $P$-{point} if each $f$ in the ring $C(X)$ of continuous real-valued functions is constant on a neighborhood of $p$. It is well-known that the ring $C(X)$ is von Neumann regular ring iff each of its elements is a von Neumann regular element; in which case $X$ is called a $P$-{space}. Properties of essential $P$-spaces (which are generalizations of J.L. Kelley's door spaces) are derived with the help of the algebraic properties of $C(X)$. www.univie.ac.at /EMIS/journals/CMUC/cmuc0403/abs/abuosba.htm   (239 words)

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