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Topic: Noetherian topological space

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	NationMaster - Encyclopedia: Glossary of scheme theory
		In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...
		Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.
		Unfortunately, while it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false.
	www.nationmaster.com /encyclopedia/Glossary-of-scheme-theory (2609 words)

	PlanetMath: Noetherian module
		There is also a notion of Noetherian for rings: a ring is left Noetherian if it is a left Noetherian module over itself; similarly for right Noetherian.
		Finally, there is a notion of Noetherian topological space which is vaguely similar in intent.
		This is version 16 of Noetherian module, born on 2001-10-15, modified 2006-09-13.
	www.planetmath.org /encyclopedia/NoetherianModule.html (262 words)

	PlanetMath: Noetherian topological space
		As a first example, note that all finite topological spaces are Noetherian.
		Cross-references: radical ideals, one-to-one correspondence, chain, subset, ideal, properties, Zariski topology, field, prime spectrum, hereditarily, compact, subspace, Hausdorff topological space, quasi-compact, compactness, noetherian, finite, integer, sequence, closed subsets, descending chain condition, topological space
		This is version 15 of Noetherian topological space, born on 2002-09-17, modified 2007-01-11.
	www.planetmath.org /encyclopedia/NoetherianTopologicalSpace.html (177 words)

	topological space
		The category of all topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in all mathematics.
		A linear graph is a topological space that generalises many of the geometric aspects of graphss 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.
	www.fact-library.com /topological_space.html (1951 words)

	Springer Online Reference Works
		An affine Noetherian scheme is precisely the spectrum of a Noetherian ring.
		A quasi-compact locally Noetherian scheme is a Noetherian scheme.
		An example of a Noetherian scheme is a scheme of finite type over a field (an algebraic variety) or over any Noetherian ring.
	eom.springer.de /N/n066860.htm (112 words)

	PlanetMath: alternative characterizations of Noetherian topological spaces
		"alternative characterizations of Noetherian topological spaces" is owned by yark.
		Cross-references: compact, subset, maximal element, minimal element, open subsets, closed subsets, Noetherian topological space, topological space
		This is version 7 of alternative characterizations of Noetherian topological spaces, born on 2004-03-19, modified 2006-07-25.
	www.planetmath.org /encyclopedia/AlternateCharacterizationsOfTheNoetherianCondition.html (102 words)

	PlanetMath: compact
		with its subspace topology is a compact topological space.
		Note: Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space.
		Since some authors require compact spaces to be Hausdorff and others don't, it is apropriate to mention this fact in the article.
	planetmath.org /encyclopedia/Compact.html (267 words)

	Proper map - Wikipedia, the free encyclopedia
		In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact.
		Thus, a topological space is compact if and only if the constant map to a point is proper.
		are closed maps of the underlying topological spaces.
	www.wikipedia.org /wiki/Proper_morphism (485 words)

	Journal of Formalized Mathematics, Index of MML Identifiers
		On the Lattice of Subalgebras of a Universal Algebra.
		Subspaces and Cosets of Subspaces in Vector Space.
		On the Lattice of Subspaces of a Vector Space.
	www.mizar.org /JFM/mmlident.html (2155 words)

	Abelian category - Wikipedia, the free encyclopedia
		If K is a commutative noetherian ring, then the category of finitely generated modules over K is abelian.
		As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over k.
		If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category.
	www.wikipedia.org /wiki/Abelian_category (943 words)

	PlanetMath: alternative characterizations of Noetherian topological spaces, proof of
		"proof of alternative characterizations of Noetherian topological spaces" is owned by yark.
		Cross-references: unions, subspace topology, stationary, subcover, finite, open cover, sequence, proper subset, strictly, infinite, bijective, map, compact, subset, maximal element, minimal element, open subsets, closed subsets, topological space
		This is version 9 of proof of alternative characterizations of Noetherian topological spaces, born on 2005-07-27, modified 2006-09-14.
	www.planetmath.org /encyclopedia/ProofOfAlternateCharacterizationsOfTheNoetherianCondition2.html (304 words)

	Noetherian (Site not responding. Last check: 2007-10-26)
		noetherian; although this is not related in a simple way to the property for rings, the definition is based on an ascending chain condition.
		site can also be noetherian; this is a generalization of the notion of noetherian for topological space.
		Noetherian rings (and by extension most other uses of the word noetherian) are named after Emmy Noether (see
	www.objectsspace.com /encyclopedia/mathematics/entries/16/Noetherian/Noetherian.html (284 words)

	The Dimension of a Space
		A space is reducible iff two nonempty open sets do not intersect, iff two proper closed sets cover the space.
		A closed subspace of a noetherian space is noetherian.
		Therefore T is noetherian, yet its dimension is infinite.
	www.mathreference.com /top-dim,intro.html (1480 words)

	noetherian ring
		In mathematics, a ring is called Noetherian if, intuitively speaking, it is not "too large" as expressed by a certain finiteness condition on its idealss.
		Noetherian rings are named after the mathematician Emmy Noether, who developed much of their theory.
		An example of a ring that's not Noetherian is a ring of polynomials in infinitely many variables: the ideal generated by these variables cannot be finitely generated.
	www.fact-library.com /noetherian_ring.html (303 words)

	[No title]
		The tangent space to the parameter space at $\alpha$ can be identified with the space of skew-symmetric bilinear forms (in the usual additive sense) $\gamma$ on $H$: a tangent vector $\gamma$ represents the 1-jet $\alpha (1 + t \gamma + \dots)$ of a curve.
		Contrary to the case of generic $q$, where such spaces were studied at the level of their noncommutative algebras of functions, see Lakshmibai-Reshetikhin \cite{lr}, Soibelman \cite{soil} and Chari-Pressley \cite{chari}, our root-of-unity spaces are honest topological spaces, ringed with rather small sheaves of noncommutative algebras.
		Since the cohomology of a noetherian topological space with constant coefficients is always zero in all degrees higher than zero, the 1-cocycle $\{c_{ij}\}$ is a coboundary and thus, $\{f_{ij}\}$ is equivalent to the class of $ \{X_i^d X_j^{-d}\}$, which is the class of $\Q(d)$ in $\Pic \np^n_\epsilon$.
	math.berkeley.edu /~alanw/Semiquantum.tex (5285 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		The vector space $V$ is an $R-$module by the natural action of the endomorphisms.
		After dualization we are even able to extend our notion of ``space'' in the sense that we can consider more general rings and regard them as dual objects of some generalized ``spaces''.
		The topological space consist of the element $[\{0\}]$ and the elements $[(p)]$ where $p$ takes every prime number.
	www.math.uni-mannheim.de /~schlich/preprints/algeo.tex (12290 words)

	Glossary of scheme theory - Wikipedia, the free encyclopedia
		For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme.
		A scheme S is a locally ringed space, so a fortiori a topological space, but the meaning of point of S are threefold:
		We can require that the scheme is covered by affine open sets whose rings of coordinates have the property P in question.
	en.wikipedia.org /wiki/Glossary_of_scheme_theory (865 words)

	Ivan Gotchev
		A topological space X is s -compact [2] if every sequentially open cover of X has a finite subcover.
		Every Noetherian topological space is s -compact and thus it is sequentially compact.
		A topological space X is called irreducible [1] if the intersection of any finite number of open nonempty sets in X is nonempty.
	www.utm.edu /staff/jschomme/topology/c/a/a/h/24.htm (452 words)

	NDSU Department of Mathematics: Colloquium
		Abstract: The Rees algebra of an ideal I in a commutative Noetherian ring A is defined as the graded subring R=A[It] of the ring of polynomials A[t].
		From there, I will generalize to stratified spaces - spaces that are not quite manifolds, but which possess sets of singularities that can be filtered into manifold strata.
		Topologically the compact orientable surfaces are completely classified as the surfaces of genus g>=0 (a sphere with g handles).
	math.ndsu.nodak.edu /colloquium (1407 words)

	1997
		Mazur, An "infinite fern" in the universal deformation space of Galois representations, 155-193.
		These spaces are generated by a function $\Psi:T\times R^{2}\rightarrow R_{+}$ such that $\Psi(\cdot,u)$ is a $\Sigma$-measurable function for any $u \in R^{2}$ and $\Psi(t,\cdot)$ is a homogeneous, concave function vanishing at zero and by a couple of Banach function lattices $E_{1}$ and $E_{2}$ over a nonatomic measure space \po.
		It is proved that, in contrast to the case of $H^p$-spaces, the space $\Smir$ is not isomorphic to the Smirnov class of holomorphic functions on the unit disc.
	www.imub.ub.es /collect/1997.html (5244 words)

	AMCA: On Sequential Properties of Noetherian Topological Spaces by Ivan Gotchev (Site not responding. Last check: 2007-10-26)
		A topological space X is s-compact [2] if every sequentially open cover of X has a finite subcover.
		A topological space X is called irreducible [1] if the intersection of any finite number of open nonempty sets in X is nonempty.
		Let X be a Noetherian topological space and P be the set of all closed irreducible subsets of X, ordered by inclusion.
	at.yorku.ca /cgi-bin/amca/caah-24 (497 words)

	[No title]
		X = Spec(R) is a noetherian topological space, since the closed subsets correspond to ideals.
		Proposition Every closed subset in a noetherian topological space can be uniquely decomposed as an irredundant union of irreducible subspaces.
		be a scheme that is a noetherian topological space.
	odin.mdacc.tmc.edu /~krc/agathos/dimen.html (969 words)

	Dimension in mathematics - Information Technology Services (Site not responding. Last check: 2007-10-26)
		in a vector space the dimension is the length of a maximal nested family of subspaces.
		This generalizes in algebraioc geometry to the definition of the dimension of a noetherian space.
		I think a noetherian space is one in which descending chains of closed sets are always finite.
	www.physicsforums.com /archive/forum/t-41627_Dimension_in_mathematics.html (2043 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		In addition, under suitable compactness conditions they are preserved under maps of the underlying spaces and have finite dimensional cohomology spaces.
		Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifolds and the structure sheaf of a Noetherian scheme from algebraic geometry.
		In case the ring R is Noetherian, coherent sheaves correspond exactly to finitely generated modules.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Coherent_sheaf (794 words)

	[No title]
		A noetherian space is a finite union of irreducible closed subspaces.
		Let $A$ be a noetherian ring, $I$ an ideal of $A$, and $M$ a finitely generated $A$-module.
		If $A$ is noetherian and $I$ an ideal of $A$, then if $x$ is not a zero divisor in $A$, neither is $\hat{x}$ in the $I$-adic completion $\hat{A}$.
	www.math.utah.edu /~opstall/writings/atmacsol.tex (8148 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		R[[X]] is actually a topological ring so that these infinite sums are well-defined and convergent.
		This topological ring is the ring of formal power series over R and is denoted by R[[X]].
		R[[X]] is an associative algebra over R which contains the ring R[X] of polynomials over R; the polynomials correspond to the sequences which end in zeros.
	www.informationgenius.com /encyclopedia/f/fo/formal_power_series.html (1281 words)

	register-forap-mat-html
		Lang, H.; 1982: On sums of subspaces in topological vector spaces and an application in theoretical tomography.
		Ellingsrud, Geir; 1983: On the representation afforded by the space of regular differentials of a group acting freely on a curve in characteristic P. Björner, Anders; 1983: On matroids, groups and exchange languages.
		Andersson, Lars; 1986: On the space of asymptotically Euclidean metrics.
	www.math.su.se /matematik/forskning/forappgamla.html (4334 words)

	Darin Stephenson \| Hope College \| Department of Mathematics
		In order to understand the proper generalizations of commutative concepts to the noncommutative case, it is often necessary to rephrase geometric results in terms of the appropriate categories.
		Given a noetherian graded algebra R, we define the category Grmod R whose objects are graded R-modules, with morphisms being defined as graded R-module homomorphisms of degree 0.
		We consider the quotient category Tails R = Grmod R / Fdim R to be the category of `representations on a noncommutative space X.' Unlike the commutative case, the space X does not necessarily exist as a topological space.
	math.hope.edu /stephenson/index-research.html (171 words)

	[No title]
		Introduction\endheading There is a class of theorems that characterize certain structures by their basic topological properties.
		Recall that a topological space is {\it Noetherian\/} if it has the descending chain condition on closed subsets.
		Complex manifolds\endheading Let $X$ be a (reduced, Hausdorff) compact complex analytic space, and consider the topology $An$ on $X^n$ whose closed sets are the closed analytic subvarieties of $X^n$.
	www.ams.org /bull/pre-1996-data/199328-2/hrushovski.tex (4243 words)

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