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PlanetMath: first axiom of countability A space that satisfies the first axiom of countability is said to be first-countable. "first axiom of countability" is owned by drini. This is version 2 of first axiom of countability, born on 2002-02-18, modified 2004-03-29. planetmath.org /encyclopedia/FirstCountable2.html   (67 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-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: 2007-10-12) First countable means that every point has a COUNTABLE local base. 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. First, tell your reader exactly which countable collection of open sets you're going to show is a base. www.math.niu.edu /~rusin/known-math/01_incoming/1st_countable   (646 words)

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)

First and Second Countable   (Site not responding. Last check: 2007-10-12) If every point p in a topological space has a countable base at p, the base is first countable. 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. www.mathreference.com /top,12cnt.html   (490 words)

Annals of Mathematics, II. Series, Vol. 152, No. 1, pp. 207-257, 2000 First of all, one defines the key notions of diverse and diffuse families of leaves over an $na$-chain $\overline{\cal M}$ of length $n$ (a leaf is a chain $\overline{\cal N}$ of length $n+1$ extending $\overline{\cal M}$ up to $\overline{\cal M}$-isomorphism). When examining why $T$ could fail to be locally tt over a chain, one realizes that a failure over a chain of length $n$ implies the existence of a diverse family of leaves of continuum size over some $na$-chain of length $n$, and a trivial failure yields in this way a diffuse family. A clear introduction first summarizes the history of the spectrum problem from Löwenheim-Skolem to Shelah, and reports Shelah's work; then it explains the genesis of this paper, as well as the plan and the basic lines of the proof. www.maths.tcd.ie /EMIS/journals/Annals/152_1/5.html   (1049 words)

[No title] A topological space $X$ is called weakly first countable, if for every point $x$ there is a countable family $\{C_n^x \mid n\in\omega\}$ such that $x\in C_{n+1}^x \subseteq C_n^x $ and such that $U \subset X$ is open iff for each $x \in U$ some $C_n^x$ is contained in $U$. This weakening of first countability is due to A. A combinatorial statement concerning ideals of countable subsets of $\omega_1$ is introduced and proved to be consistent with the Continuum Hypothesis. www.cs.bgu.ac.il /~abraham/math.html   (2171 words)

[No title] Thus the general form of a closed singular term in L that is not a name will be f(t1,...,tn), where f is an n-place function sign and t1,...,tn are closed terms. In any countable first-order language L, every replete set can be made true in a (naturally definable) countable model for L. Proof. A countable model for L consists of a countable domain of individuals, with specifications of denotations for names in L, mappings for function signs in L, and extensions for primitive predicates in L. Let (be a replete set (in L). people.cohums.ohio-state.edu /tennant9/clean_classical_completeness.doc   (1168 words)

Countably/Sequentially Compact Assume a space s is second countable, and let u be countably compact in s. Countable compactness is equivalent to the statement that every countable collection of closed sets with the finite intersection property has a nonempty intersection. Next assume s is countably compact and first countable. www.mathreference.com /top-cs,count.html   (582 words)

Ordinal number   (Site not responding. Last check: 2007-10-12) Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. (See How to name numbers.) ---- In mathematics, ordinal numbers are an extension of the natural numbers to accommodate infinite sequences, introduced by Georg Cantor in 1897. The first transfinite Ordinal is ω, the set of all natural numbers. as a limit of countably many smaller ordinals) and are called limit ordinals; the other ordinals are the successor ordinals. ordinal-number.iqnaut.net   (1042 words)

Spring 96 Abstracts The first is that if 0 is in the interior of the convex hull of the recurrent rotation vectors for an area preserving diffeomorphism f isotopic to the identity then f has a fixed point of positive index. H. Furstenberg was the first to establish the existence of relationships between recurrence, IP sets, and idempotents in the enveloping semigroup, and the first author has proven that the closure of the set of idempotents coincides with the IP cluster points. The model is investigated first numerically, and then simpler models of the Poincaré return map (based on our observations of behavior of the flow) are explored from a rigorous and topological standpoint. math.tntech.edu /spring-top/spring96-abstracts.html   (11944 words)

[No title]   (Site not responding. Last check: 2007-10-12) Clearly Q and N are first countable as are Q times Q and Q times N. Generally, an arbitrary quotient space of a first countable space need not be first countable. Similarly, Q times N is first countable as are (Q smash Q) times N and Q times (Q smash N). It seems to me that everything in sight is first countable and therefore is a k space. www.lehigh.edu /~dmd1/mc25   (266 words)

Untitled Document The question doesn't make any sense because water is non countable. - A or an is used to introduce a noun when it is mentioned for the first time in a piece of writing. There are situations, however, when the newspaper must determine whether the public's safety is jeopardized by knowing the truth. www.freewebtown.com /cmdpro/PinarTasci/further.htm   (343 words)

Set_Theory_Beginning Any infinite subset of the natural numbers or the integers is countable. Thus b is a member of s but b not equal to f(k) for any k, so f is not "onto". Hence S is not a countable set, and consequently neither is [0,1]. www.humboldt.edu /~mef2/book/Set_Theory_Begin.htm   (426 words)

[No title]   (Site not responding. Last check: 2007-10-12) 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)

First Countable Spaces   (Site not responding. Last check: 2007-10-12) If is a countable base for and, consider which is a countable family of neighbourhoods of p. An uncountable discrete space is first countable (since metrizable), yet is not completely separable. Then is a countable family of open sets and is a base for. at.yorku.ca /course/atlas3/node9.html   (177 words)

hair loss, hair removal, hair growth, clinical studies. The regrowth of hair was first noticed on some participants as early as two months into the treatment. From the photos taken monthly during the course of the study, a pattern of regrowth was observed in those participants who demonstrated a significant increase in hair counts over the five months. The first countable hairs were seen after 10-12 weeks of treatment and were first detected on the crown or vertex of the head. www.nisim.com /page.help   (2800 words)

Countable vs Nouns that are countable in English may be non-countable in other languages, including your own, and vice versa. While the issue of countability is too complicated for us to categorize each noun absolutely, we can still describe some general patterns. (For example, "You're missing three of the homeworks from the first part of the course.") Because this usage is not firmly established and is likely to be considered nonstandard, you should check with your instructor before using it in writing. www.sabri.org /count-2.htm   (919 words)

Atlas: A first countable linearly Lindelof not Lindelof space by Oleg Pavlov   (Site not responding. Last check: 2007-10-12) A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. 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 # cate-01. atlas-conferences.com /cgi-bin/abstract/cate-01   (187 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. All superclasses of closed unbounded classes are stationary and stationary classes are unbounded, but there are stationary classes which are not closed and there are stationary classes which have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). 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)

My Discoveries   (Site not responding. Last check: 2007-10-12) First, if there is in fact a finite formula that uniquely describes the real, then any particular piece of information must be computably extractable from it. First, the first w countable ordinals map onto the aleph_0 computable real numbers. First of all, the constructable universe is the set of all sets generable from the ZF axioms (this is distinct from the intuitionist concept of constructable). www.wall.org /~aron/disc.html   (1910 words)

The Official Site of the New York Islanders - Post-Game Article   (Site not responding. Last check: 2007-10-12) Kristian Huselius scored the first countable goal of the contest six minutes in when he beat DiPietro between the wickets from a sharp angle. It was the first of two goals that Parrish played a big part in, but didn't get a point to show for it. The Islanders' penalty killing streak was extended to 34 as they killed all six of Florida's power plays...After taking 12 shots on goal in the first two periods combined, the Islanders took another 12 in the first half of the third period... www.newyorkislanders.com /pressbox/postgame.asp?id=570   (737 words)

Atlas: First countable extensions of regular spaces by Petr Simon   (Site not responding. Last check: 2007-10-12) We will present a technique, which allows to find a feebly compact, (pseudocompact, resp.) envelope of a given space, which preserves first countability or even Mooreness of the input space. Every locally feebly compact regular space X can be embedded as a dense open subspace in a feebly compact regular space Y which is first countable at every point from Y\X. Every separable, locally feebly compact Moore space embeds as an open dense set in a feebly compact Moore space. The first two theorems answer Stephenson's questions 23 and 25 from [S] and significantly strengthen [T, Theorem 2.2]. atlas-conferences.com /cgi-bin/abstract/cajv-22   (378 words)

DC MetaData for: A better framework for first countable spaces   (Site not responding. Last check: 2007-10-12) Abstract: In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Several applications of first countability in a broader context than the usual one of topological spaces are studied. Keywords: First axiom of countability, second axiom of countability, countably compact, sequentially compact, sequentially complete, continous convergence, sequentially continous, semiuniform convergence spaces, convergence spaces, filter spaces, topological spaces, uniform spaces, bicoreflective subconstruct, cartesian closedness www.math.fu-berlin.de /publ/preprints/2002/Ab-A-02-01.html   (138 words)

English Grammar Articles,Definite Article,Indefinite,Second Language Exercise,Lesson A or an is used to introduce a noun when it is mentioned for the first time in a piece of writing. A, an, and the can all be used to indicate that a noun refers to the whole class to which individual countable nouns belong. While some nouns combine with one article or the other based on whether they are countable or non countable, others simply never take either article. www.learn4good.com /languages/evrd_grammar/articles.htm   (762 words)

Springer Online Reference Works Sequential spaces form a coreflective subcategory (see Reflective subcategory) of the category of all topological spaces; the coreflection is obtained by retopologizing an arbitrary space with the topology in which a subset is closed if and only if it is closed under limits of sequences (in the original topology). Spaces which satisfy the first axiom of countability are always sequential (indeed, they are Fréchet–Urysohn spaces), and the sequential spaces form the smallest coreflective subcategory containing all first-countable spaces. For this reason, many topological results which are traditionally proved for first-countable spaces can readily be extended to sequential spaces. eom.springer.de /s/s084620.htm   (134 words)

Topological space   (Site not responding. Last check: 2007-10-12) The sets in T are the open sets, and their complements in X are the closed sets. The first axiom is redundant and just included for clarity. A space is countably Compact if every countable open cover has a finite subcover. topological-space.iqnaut.net   (2393 words)

Large Numbers at MROB , "epsilon-null", to be the first ordinal infinity that could not be expressed with a finite number of omegas and/or integers combined with addition, multiplication, and exponents. There is actually no particular reason why Cantor had to do this, except that he did not consider using a hyper4 operator. Whereas the first three infinities can be thought of as counting the number of integers, points, and curves in 2-d space, 2 home.earthlink.net /~mrob/pub/largenum-4.html   (2727 words)

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