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Topic: Second countable

	Second-countable space - Wikipedia, the free encyclopedia
		In topology, a second-countable space is a topological space satisfying the "second axiom of countability".
		Although the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates.
		In second-countable spaces—as in metric spaces—compactness, sequential compactness, and countable compactness are all equivalent properties.
	en.wikipedia.org /wiki/Second-countable_space (409 words)

	PlanetMath: second countable
		line is both Lindelöf and separable, but not second countable.
		See Also: separable, Lindelöf, every second countable space is separable, Lindelöf theorem, Urysohn metrization theorem, first axiom of countability
		This is version 12 of second countable, born on 2002-01-01, modified 2005-02-11.
	planetmath.org /encyclopedia/SecondCountable.html (119 words)

	Topology glossary
		The first part deals with general concepts, and the second part lists types of topological spaces defined in terms of these concepts.
		A space is separable if it has a countable dense subset.
		A space is second-countable if it has a countable base for its topology.
	www.ebroadcast.com.au /lookup/encyclopedia/lo/Local_base.html (1004 words)

	First and Second Countable (Site not responding. Last check: 2007-11-03)
		This is an indispensable property of first countable, and it is used in various proofs.
		Restrict radii to rational values, and the balls centered at p are countable.
		A topology is second countable if it has a second countable base.
	www.mathreference.com /top,12cnt.html (490 words)

	First-countable space - Wikipedia, the free encyclopedia
		In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".
		An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).
		In first-countable spaces, sequential compactness and countable compactness are equivalent properties.
	en.wikipedia.org /wiki/First-countable_space (325 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		A first countable space need not be second countable --- consider any uncountable discrete space.
		As for what you want to prove, namely that for countable topological spaces, first countability implies second countability, you want to be a bit more explicit in your proof.
		[Each point has a countable local base (these bases may not be unique, so make sure you don't refer to any of them as "the local base at x" until you've fixed a choice of them in your proof), there are countably many points, a countable union of countable sets is countable,.
	www.math.niu.edu /~rusin/known-math/01_incoming/1st_countable (646 words)

	Real line
		Second, the real numbers can be turned into a metric space by using the metric given by the absolute value: d(
		It is paracompact and second countable as well as contractible and locally compact.
		It also has a standard differentiable structure on it, making it a differentiable manifold.
	www.ebroadcast.com.au /lookup/encyclopedia/re/Real_line.html (405 words)

	PlanetMath: second countable
		A topological space is said to be second countable if it has a countable basis.
		It can be shown that a second countable space is both Lindelöf and separable, although the converses fail.
		second axiom of countability, completely separable, perfectly separable
	planetmath.org /encyclopedia/SecondAxiomOfCountability.html (119 words)

	Ordinal number Summary
		A runner who comes in first comes in ahead of anyone else, and that is what is most notable about the event, not that one runner has crossed the line.
		Likewise, someone coming in second "follows," and that, too, is something which can be noted without consciously counting the two runners.
		in the name, this ordinal is countable), which is the smallest ordinal which cannot in any way be represented by a computable function (this can be made rigorous, of course).
	www.bookrags.com /Ordinal_number (4728 words)

	Urysonh's Metrization Criteria
		Every metric space is normal, but may not be second countable, as shown by the descrete topology on an uncountable set.
		Each isolated point is open, and is the union of base sets, hence there is a base open set for each point, which contradicts second countable.
		Since p is the countable product of metric spaces it is metrizable, as demonstrated in the previous section.
	www.mathreference.com /top-ms,umc.html (1002 words)

	[No title]
		Find a countable > topological space which is not second-countable (and thus not first-countable).
		All points except (0,0) are open; as for (0,0), a neighborhood of that point is a set U such that, for all but finitely many choices of m, the sets I_m = { n
		This _does_ give a topology and it's not hard to prove that there is no countable local basis at (0,0), hence the space is not 1st-countable.
	www.math.niu.edu /~rusin/known-math/95/1stcountable (190 words)

	The Hierarchy of Borel Universal Sets
		-universal set parametrised by a second countable space is metrisable’ and ‘every compact, first countable space with a
		Second, when looking for absolute theorems, our parametrising space will need to have a strong property hereditarily.
		Each section is further divided in two, with the first part dealing with general spaces, and the second the special case of compact spaces.
	pear.math.pitt.edu /mathzilla/Examples/DynHieruni.xml (1264 words)

	Topology MAT 530
		The Baire theorem states that in a complete metric space, the intersection of countably many open dense sets is dense ("dense" means that the closure is the whole space).
		In other words, a countably family of continuous functions is enough to separate any point from any closed set not containing this point.
		The second countability axiom states that there is a countable basis.
	www.math.sunysb.edu /~azinger/mat530/fall04/index.htm (2907 words)

	second countable metric Spaces
		Let X be a second countable metric spaces(X has a countable dense
		Because X is 2nd countable metric, it has at most countably many
		The same also for any other countable dense set of X. When S infinite, X may have non-homeomorphic countable dense subsets.
	sci4um.com /about6324.html (488 words)

	Topological manifold - Wikipedia, the free encyclopedia
		In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable.
		However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components.
		For example, for differentiable manifolds the transition maps are required to be a diffeomorphisms.
	en.wikipedia.org /wiki/Topological_manifold (1446 words)

	S.O.S. Mathematics CyberBoard :: View topic - Second Countable
		A space is called second countable if it has a countable open base.
		are rational is a countable open base, since every open interval and hence every open set can be made from unions of these.
		Show a second countable space is compact if and only if every countable open cover has a finite subcover.
	www.sosmath.com /CBB/viewtopic.php?t=21831&sid=f12e09a1b19c634504acd1474f77ea8f (243 words)

	Lindelof + Metrizable ==> Second Countable
		countable subcollection, it may be the case that x_i is outside of U for
		your space is second countable (because it ain't!).
		Since X is Lindelof, there is a countable subcollection, let's call it
	sci4um.com /about4546.html (574 words)

	Infinite Ink: The Continuum Hypothesis by Nancy McGough
		The lowest level is called "countable infinity" and higher levels are called "uncountable infinities." The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set.
		A topological space X is a continuous curve iff X is a compact Hausdorff space which is second countable, connected, and locally connected.
		But, within this system every set of reals is either countable or has the cardinality of all the reals so the first three of the six versions of CH listed in section 1.1 hold.
	www.ii.com /math/ch (4563 words)

	School of Mathematics (Site not responding. Last check: 2007-11-03)
		General Topology: Neighbourhoods, first countable, inadequacy of sequences, second- countable, (relationship to separability), continuity of functions at points, product topology (weak topology for continuous projections).
		Objectives: This course aims to introduce general techniques used widely in analysis (and other branches of mathematics) and to treat a few topics that are active areas of research.
		Translated from the second Russian edition and edited by Richard A. Silverman.
	www.maths.tcd.ie /pub/official/Courses06-07/321.html (195 words)

	ONT Re: Differential Geometry for Engineers
		Let M be a second countable, Hausdorff topological space.
		second definition of a manifold can now be given:
		is a second countable Hausdorff topological space, and !A! is a maximal C^oo atlas.
	suo.ieee.org /ontology/msg04060.html (333 words)

	IngentaConnect Striking differences between ZF and ZF+weak choice in view of met... (Site not responding. Last check: 2007-11-03)
		Well-known properties held by metric spaces with the countable chain condition (ccc) such as separability, second countability, Lindelöf, paracompactness may fail in the absence of the axiom of choice AC.
		In the long catalogue of weak choice principles, see [10], we find those which are sufficient (and in some cases necessary) to establish well known equivalences in ZFC between the statements CCMX, where CCMX stands for: ccc metric spaces have the property X, X
		We close this gap by showing that the answer to the above question is affirmative.
	www.ingentaconnect.com /content/nisc/qm/2002/00000025/00000004/art00001 (272 words)

	Michi’s blog
		This is the second weekend in a row spent to more or less large part in the office, working with the product structures on cohomology.
		A paper recently up on arXiv details the errors committed by an author of a paper in Non-Linear Analysis, who, by ignoring basic conditions of theorems manages to prove most of mathematics and substantial parts of physics inconsistent.
		This is the second insufficiently reviewed paper at that Journal causing some sort of waves spreading as far as to me so far.
	blog.mikael.johanssons.org (5704 words)

	Re: Lindelof + Metrizable ==> Second Countable (Site not responding. Last check: 2007-11-03)
		Subject: Re: Lindelof + Metrizable ==> Second Countable
		Since X is Lindelof, there is a countable subcollection, let's call it { B(x_n ; r_n) }
		However, what you've proved is still true: there's a countable set of open balls whose union is X; in fact, you can take one single ball, namely B(x,2) for some in X. However, you can't deduce from this that your space is second countable (because it ain't!).
	www.newsfeeds.com /archive/sci-math/msg38816.html (195 words)

	The Hierarchy of Borel Universal Sets by Paul M. Gartside and Joseph Tzan Hang Lo (Site not responding. Last check: 2007-11-03)
		There is an example of a space that is not second countable, but with a G
		If there exist Q-sets or under CH, there is a compact, first countable non-metrisable space with a \Sigma
		-universal set parametrised by a second countable space is metrisable" and "every compact, first countable space with a \Sigma
	at.yorku.ca /i/d/e/b/27.htm (293 words)

	Amazon.com: Continuous Bounded Cohomology of Locally Compact Groups: Books: Nicolas Monod,J. M. Morel,F. Takens,J.M. ... (Site not responding. Last check: 2007-11-03)
		Key Phrases: compact second countable group, augmented resolution, bounded cohomology, Banach G-module, Banach H-module (more...
		Key Phrases - SIPs: compact second countable group, augmented resolution, bounded cohomology, second countable groups, finite invariant measure (more)
		compact second countable group, augmented resolution, bounded cohomology, second countable groups, finite invariant measure, irreducible lattices, coefficient modules, continuous cohomology, rough actions, finite index subgroups, amenable actions, finite orbit, closed normal subgroup, compactly generated, conjugation action, bounded orbits, isometric isomorphism, spectral sequence, locally compact group, comparison map, non elementary, topological isomorphisms, closed subgroup, topological group
	www.amazon.com /Continuous-Bounded-Cohomology-Locally-Compact/dp/3540420541 (836 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		, there is a countable collection of open neighborhoods of
		-compact if it can be covered by countably many open sets each of which is contained in a compact subspace.
		is said to be a Baire space if the following condition holds: Given any countable collection
	www.math.rochester.edu /people/grads/rld/topology/defs.html (137 words)

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