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Topic: Category of topological spaces

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	Topological space - Wikipedia, the free encyclopedia
		Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity.
		The category of topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics.
		Sierpiński space is the simplest non-trivial, non-discrete topological space.
	en.wikipedia.org /wiki/Topological_space (1777 words)

	Category of topological spaces - Wikipedia, the free encyclopedia
		In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps.
		The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.
		The coproduct is given by the disjoint union of topological spaces.
	en.wikipedia.org /wiki/Category_of_topological_spaces (535 words)

	file_nav_name Encyclopedia Index
		In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing al...
		In mathematics, a monoidal category (or tensor category) is a category equipped with a binary "tensor" functor and...
		Categorial grammar is a term used for a family of formalisms in natural language syntax motivated by the principle of...
	www.brainyencyclopedia.com /topics/category.html (8922 words)

	PlanetMath: category of pointed topological spaces
		This yields a functor from the category of pointed topological spaces to the category of groups.
		"category of pointed topological spaces" is owned by mathcam.
		This is version 3 of category of pointed topological spaces, born on 2003-10-15, modified 2004-01-24.
	planetmath.org /encyclopedia/CategoryOfPointedTopologicalSpaces.html (193 words)

	Category theory - Wikibooks, collection of open-content textbooks
		The notion of category being established as that which gives precision to the concept of domain of mathematical discourse, the formalization of the precise notion corresponding to the intuitive idea of the interrelation or connection between different domains is now considered.
		It is a functor, however, from the category of groups and surjective homomorphisms to the category of groups and all homomorphisms, because a surjective homomorphism does not necessarily map the centre surjectively.
		Further, a not unimportant purpose of the language of categories and categorical reasoning is to identify within a given argument that part which is trivial and separate it from the part which is deep and proper to the particular context.
	en.wikibooks.org /wiki/Category_theory (4088 words)

	[No title]
		The classification space of M is denoted class(M), and is defined to be B(we M), the classifying space of the category of weak equivalen* *ces of M.
		By space we always mean "simplicial set" unless otherwise indicate* d; the category* of spaces is denoted by S. Particular examples of spaces which we shall need are [n], the standard n-simplex, _[n], the boundary of the standard n-simplex, and k[n], the boundary of the standard n-simplex with the k-th face removed.
		Ho V on homotopy categories is an equiva- lence of categories, and 2.
	hopf.math.purdue.edu /Rezk/rezk-ho-models.txt (12556 words)

	Category Theory
		Category theory is both an interesting object of philosophical study, and a potentially powerful formal tool for philosophical investigations of concepts such as space, system, and even truth.
		Category theory is, in this sense, the legitimate heir of the Dedekind-Hilbert-Noether-Bourbaki tradition, with its emphasis on the axiomatic method and algebraic structures.
		From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics.
	plato.stanford.edu /entries/category-theory (11786 words)

	[No title]
		Perhaps it is not so obvious, however, that injective spaces are also closed under the formation of function spaces, once the space of continuous functions is given the right topology; indeed the category of injective spaces and continuous functions is a cartesian closed category.
		Many of the properties of these spaces are provable once the spaces can be characterized as a kind of complete lattice with an appropriate, uniquely determined topology; continuity of functions then comes down to preservation of sups of directed subsets of the lattice.
		As the function space between two powersets proves to be a retract of a suitable powerset, the desired properties follow.
	www.stanford.edu /~sommer/Scott.html (287 words)

	convergence spaces (Site not responding. Last check: 2007-10-31)
		Convergence spaces are for topological spaces like complex numbers are for real numbers; where some topological problems fail to find their solutions in topologies, they will, however, in convergences.
		The class of sequential topological spaces is of particular interest, on one hand because it is exactly the class of spaces for which sequences suffice to describe the topology, on the other hand because this is exactly the class of topological quotient of metrizable spaces.
		The subclass of sequential spaces that are stable under subspaces is that of Fréchet-Urysohn spaces.
	www.cs.georgiasouthern.edu /faculty/mynard_f/convergences.htm (3488 words)

	An Introduction to Algebraic Topology (Site not responding. Last check: 2007-10-31)
		A category is defined by its objects and morphisms.
		The problem of classification can thus be re-stated as the study of this category; can one find an algebraic description of this category that does not refer to the underlying notions of topological spaces and continuous maps.
		Thus the study will be much simpler than the study of “all” topological spaces and “all” continuous maps; as we shall see we can (essentially) construct these spaces out of building blocks as also construct maps out of simple incidence and collapsing maps.
	www.imsc.res.in /~kapil/geometry/topol/intro.html (925 words)

	CategoryTheory - The Haskell Wiki (Site not responding. Last check: 2007-10-31)
		Category theory is the mathematical study of categories.
		Many objects of interest in mathematics happen to congregate in categories, and many objects of interest in mathematics also happen to be categories themselves.
		Grph, the category of graphs and graph morphisms.
	www.haskell.org /hawiki/CategoryTheory (170 words)

	PlanetMath: topological sum
		A basis for this topology consists of the union of the set of open subsets of
		Cross-references: open subsets, union, basis, continuous, inclusion maps, disjoint union, topological spaces
		This is version 3 of topological sum, born on 2004-10-05, modified 2005-02-10.
	planetmath.org /encyclopedia/TopologicalSum.html (74 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		The second one assumes that we are modeling orbispaces by topological groupoids.
		1) The usual Grothendieck topology on the category of topological spaces is not very compatible with the operation of glueing cells.
		I would suggest the following one: The underlying category is the category of compact spaces; a morphism A -> B is a cover if it is surjective (on the underlying sets).
	www.lehigh.edu /dmd1/public/www-data/gm1214 (586 words)

	ABSTRACTS (Site not responding. Last check: 2007-10-31)
		Morel and Voevodsky constructed topological realization functors from their homotopy category of schemes over $\mathbb{C}$ to the ordinary homotopy category of spaces.
		The long-term goal is to use these functors as calculational tools in the homotopy category of schemes.
		Philosophically, higher homotopy operations are connected with higher homotopies, which arise when one tries to lift a commutative diagram in the homotopy category to topological spaces (e.g., replacing a homotopy-associative H-space by an equivalent topological group); we shall try to make this relation more explicit.
	www.math.uchicago.edu /~mandell/seminar/abstracts-200010.html (447 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		E Schlaepfer: On Chu-spaces and group algebras To imitate the construction of a group algebra for finite groups in the case of topological Hausdorff groups the category of finite vector spaces is replaced by the -autonomous category* V of delta-balls.
		The category of topological spaces and k-continuous maps takes the place of finite sets.
		Using recent results by Kleisli, Kunzi and Rosicky, we can use V to define a group algebra in the sense that there is a (functorial) bijection between the (k-continuous) linear representations and the module structures on the group algebra.
	www.math.mcgill.ca /rags/seminar/Schlaepfer.txt (138 words)

	AMCA: A fuzzy category on the basis of the category L-TOP of L-topological spaces by Alexander Sostak
		AMCA: A fuzzy category on the basis of the category L-TOP of L-topological spaces by Alexander Sostak
		[4]) having L-kernel spaces and all mappings between the corresponding sets as potential objects and potential morphisms respectively, and L-valued subclasses of objects and morphisms determined by \omega and \mu respectively.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/e/q/18.htm (352 words)

	A fuzzification of the category of $M$-valued $L$-topological spaces by Tomasz Kubiak and Alexander P. Sostak (Site not responding. Last check: 2007-10-31)
		A fuzzification of the category of $M$-valued $L$-topological spaces by Tomasz Kubiak and Alexander P. Sostak
		A fuzzy category is a certain superstructure over an ordinary category in which "potential" objects and "potential" morphisms could be such to a certain degree.
		The aim of this paper is to introduce a fuzzy category FTOP(L, M) extending the category TOP(L, M) of M-valued L-topological spaces which in its turn is an extension of the category TOP(L) of L-fuzzy topological spaces in Kubiak-Sostak's sense.
	at.yorku.ca /i/a/a/k/06.htm (136 words)

	The Monoidal Category of Hilbert Spaces
		As we shall explain, categories of this sort are called `monoidal'.
		So, the first move in category theory is to stop focussing on ordered pairs and instead focus on cartesian products of sets.
		Category theorists therefore feel free to speak of `the' product when it exists.
	math.ucr.edu /home/baez/quantum/node4.html (2679 words)

	The Largest Topological Subcategory of Countably-based Equilogical Spaces - Menni, Simpson (ResearchIndex)
		Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed --- for example, the category of sequential spaces.
		In this paper we show that the two approaches are equivalent for a large class of objects.
		2: Topological and limit-space subcategories of countablybased equilogical spaces - Menni, Simpson - 1999
	citeseer.ist.psu.edu /154350.html (306 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		Metric spaces also benefitted from a similar treatment in Lawere's fundamental paper [3], but for the identity monad this time.
		In recent years, a unified setting emerged, which allowed for a description of these categories, along with the categories of approach and uniform spaces, as categories of so-called "lax algebras" (see [4]).
		In this talk, we will present this approach and explain how lax algebras may be related to closure spaces.
	www.math.mcgill.ca /~rags/seminar/seal.12Oct04.txt (125 words)

	Citebase - Fuzzy functions and an extension of the category L-Top of Chang-Goguen L-topological spaces (Site not responding. Last check: 2007-10-31)
		Fuzzy functions and an extension of the category L-Top of Chang-Goguen L-topological spaces
		Authors: Sostak, Alexander P. We study FTOP(L), a fuzzy category with fuzzy functions in the role of morphisms.
		Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0204139 (156 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		Our goal is to show any other category satisfying these axioms is equivalent to the category of loopless pointed matroids and strong maps.
		Schlomiuk studied the category of topological spaces and continuous mappings and found twelve axioms satisfied by this category.
		In addition, she proved that any category satisfying these twelve axioms is equivalent to the category of topological spaces and continuous mappings.
	www.umt.edu /math/Colloq/fall97/110697.html (252 words)

	Quaestiones Mathematicae - Vol. 24, No. 4 (2001)
		The class of spaces with non-idempotent stratified fuzzy interior
		convergence spaces and a first characterization, which fuzzy convergences stem
		This site is maintained by NISC SA (National Inquiry Services Centre) as an initiative to support African-published journals.
	www.ajol.info /viewarticle.php?id=8881 (118 words)

	Quaestiones Mathematicae - Vol. 25, No. 1 (2002)
		differential space that can be built up from cells and whose differential
		This concept stems from an analogue in the category of
		investigating the underlying topological space of a DW complex.
	www.ajol.info /viewarticle.php?id=8974 (120 words)

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