Fusion power - Revision history

en>Harriv: typo in publication title

2015-01-09T12:22:05Z

typo in publication title

← Older revision		Revision as of 14:22, 9 January 2015
Line 1:		Line 1:
	~~Keep~~ a ~~food journal~~ to ~~record the number of calories you're consuming~~. ~~And workout more calories~~ in ~~per day than you burn~~, ~~reduction supplement just wouldn't happen~~. ~~Eating too many calories minimizes you from losing weight~~. ~~You can write down your diet~~ to ~~assure you~~'~~re eating~~ the ~~quantity calories needed to lose diet~~.<br><br>~~To stay motivated while losing weight~~, ~~reward yourself in ways that aren't affiliated with food. As small treat sometimes can~~ be ~~good, but try~~ to ~~get out of~~ the ~~habit of smoking of taking into consideration food in exchange~~. ~~Instead~~, ~~promise yourself a whole outfit~~--~~in whole lot size!~~--~~or a vacation~~ in ~~the day spa~~.<br><br>~~It is utterly harmless without side affects~~. ~~This~~ the ~~reason as to why~~ people ~~are using it any kind~~ of ~~fear since know that it's going~~ to ~~not~~ to ~~be able~~ to ~~harm them in in any manner until and unless it's mixed with any other compound~~.<br><br>~~Start exercising. If you have to do not have enough time to do exercise (as most belonging~~ to the ~~people do not find time)~~, ~~start simply walking~~. ~~Include walk with your daily run~~-of-~~the~~-~~mill~~. ~~You~~ can ~~walk anywhere~~. ~~Is actually always strongly recommended~~ to ~~walk at least~~ a ~~mile every day and gradually add miles~~.<br><br>~~You~~ the ~~proper way through many processes~~. ~~The actual first is~~ the ~~intake~~ of ~~much calories~~ and ~~other~~ is ~~being unable~~ to ~~burn down calories then it does not turn into fat~~. ~~Your own body follows~~ a ~~normal metabolism extent~~. ~~If you take in more numbers of calories inside your body then the body becomes unable burn off down high~~. ~~Thus these unburned calories settle as fat inside one's body~~. ~~But with Raspberry ketone you~~ can ~~avoid our~~. ~~Raspberry~~ [http://~~superketoneinfo~~.~~com~~/ ~~Super Ketone Review~~] ~~helps to increase one's metabolism rate thus your fat gets burned out~~. ~~Aid in pounds reduction and reduced fat content~~ [http://~~superketoneinfo~~.~~com~~/ ~~Super Ketone Review~~] [http://~~superketoneinfo~~.com/ ~~Super Ketone Review~~] ~~in entire body~~.<br><br>~~Go public with your weight loss goals~~. ~~Telling everyone your know you'll be trying to loss weight can like a great motivator as you'll be afraid from the shame could feel this would mean succeed~~. ~~Letting others know will also prevent them from promoting fat-filled snacks when you meet out~~.<br><br>~~Turn the switch off everything as well as obtain some unwind time~~ (~~without the TV~~ ) ~~60 minutes before to be able to to bed or crib~~. ~~You can came up with a consistent ritual in this~~. ~~Throw the tv out of the bedroom~~ (~~seriously)~~. ~~Drink a soothing herb tea~~ - ~~no toddies or pills feel free~~ to.<br><br>~~After observing reviews and visiting health outlets~~, ~~it seems sensible a positive perspective~~. ~~Expense is average: $30~~-~~$100 in a supply that last 30 to180 days~~. ~~In addition~~, ~~this is in order to produce an increase in metabolism with a loss in appetite that will contribute to losing about 2~~.~~5lbs~~. ~~per week~~. ~~That amount is not considered dangerous to normal either~~. ~~Of course, weight should be lost from a safe manner~~.		== Eventually the sugar rush ends Nike Free Sale Women ==

			She spent nearly Ten years recruiting, training and counseling college graduates for large companies and enjoyed the chance to teach and mentor people at a critical time in their careers. When she'd her baby in early 2010, she decided to find new ways to engage, teach and motivate others through her other passion: fitness. <br><br>Patrons can log in and use their own digital coupon clipping system. The process is quick, easy, and it helps shoppers save your time and hardearned money.. Mr Monk, who delivers machinery to HEB Construction's work sites, said Mr Taiaroa worked for the company for about 14 years. He was popular and would take his mates diving and fishing on their own days off work. <br><br>Lots of times people are searching for news about a particular subject, and consequently the search engines will rank newly published information highly to be able to satisfy the searchers' (possible) need for recently published articles. This brings home the truth that there is a need to keep adding content to your site, and it shows that opportunities are available for sites that [http://www.trustfornature.org.au/Shared/Tasks/Favicon.asp?id=15-Nike-Free-Sale-Women Nike Free Sale Women] are regularly adding new content.Lot's of other great info is coming that I'm looking forward to sharing!I'm making the trip to SMX Advanced in Seattle. <br><br>You can find lots of funny pictures. There exists a big photos gallery from different kinds of categories. Though I have known a few people who have stopped the paid inclusion listing (and also the perclick fee) and have remained in the listings, most people totally give up of the rankings. So if you get where you want to be, be prepared to pay to remain there. <br><br>Set behind a farm and shut to the road network of this area of Lancashire, this campsite takes caravans, motorhomes and tents and it is quiet and sheltered. It forms a part of Hardhorn Breaks which also have holiday apartments services [http://www.caloundrabridgeclub.com.au/results/unprocessed/frames.asp?headeventid=28-Nike-Shox-Size-14 Nike Shox Size 14] next door. Dress well Properly fitting clothing can do two things for you. 1 Cause you to look thinner and 2 remind you that you've a belly which can keep you from eating too much. <br><br>Eventually the sugar rush ends, depression occurs, after prolonged exposure malnutrition and death result. Only Christ provides the truly nourishing and energysustaining banquet from the Bread of Life and Living Waters.. The following MUBBEG Business Breakfast will be held on Friday 11 November 2005 from 7AM 8.30AM. Professor Phillip Steele, Pro ViceChancellor (Campus Coordination) Academic Director, Berwick/Peninsula will talk about The Urban Plan for Monash University Berwick Campus.DONATE BLOODDO NOT Disregard the NEED FOR BLOOD!The Australian Red Cross Blood Service mobile unit is going to be at Berwick Secondary College, Manuka Road, Berwick on Friday 25 [http://www.roseofraphael.com.au/images/cards/large/aspmail.asp?a=110-Discount-Ray-Bans-Australia Discount Ray Bans Australia] November [http://www.costeng.com/includes/wrapper.asp?q=109-Nike-Store-Sydney Nike Store Sydney] from 9.45am 3.15pm.All faculties can now make adhoc room bookings directly in MUTTS.<ul>

			<li>[http://nanchong.tftong.cn/forum.php?mod=viewthread&tid=370816&fromuid=1193 http://nanchong.tftong.cn/forum.php?mod=viewthread&tid=370816&fromuid=1193]</li>

			<li>[http://www.3y2.cn/forum.php?mod=viewthread&tid=829379 http://www.3y2.cn/forum.php?mod=viewthread&tid=829379]</li>

			<li>[http://atpic.cn/forum.php?mod=viewthread&tid=58736 http://atpic.cn/forum.php?mod=viewthread&tid=58736]</li>

			<li>[http://forum.tag-knowledge.com/default.aspx?g=posts&t=1079219 http://forum.tag-knowledge.com/default.aspx?g=posts&t=1079219]</li>

			</ul>

			== Ugg Sko Til Barn det var en messe gjør ekte tatoveringer ==

			Konseptet og de potensielle resultatene fra den eneste tanken høres imponerende, men det er alltid en ulempe, og dette er som et thermolagringssystem . Nåværende sportsnyheter i hockey, baseball, tennis, fotball, basketball og golf. I tillegg til kostnaden av rattet Jones var fryktelig av sin vekt, pluss skipperen vekt, blir i det ekstreme akterenden av båten. <br><br>Pinnen figuren er ment å være hammeren kasteren og ordet tror har den enorme utropstegn ved siden av den. Så selv om gratis er tilgjengelig, er det aldri den beste solution.Also, hvis du bruker gratisversjonen av MSSQL 2005 vil du rett og slett ikke være i stand til å konvertere mellom databasestrukturer, som jeg nevnte tidligere du får det du betaler for; bedre . <br><br>Swaim referanser Mike Mignola Hellboy comicbook serien ikke filmene, Swaim er raske til å klargjøre som en innflytelse på både måten historien utfolder seg og hvordan den bruker folklore som en jumpingoff punkt. Dette gir en mulighet til å fjerne noen av de nevnte cluttermaybe noen hus annonser annonsering BMJ hendelser, feeds noe innhold som aren får mange klikk etc.. <br><br>Har Begge mennene syndet før Gud og andre. Daglig (Hvis ikke oftere.) Revolutionary Wannabees On The Mars. Viseministeren Faizser Mustapha som har formannskapet et reiselivsutvalg i [http://www.cobsen.com.br/contador/functions.php?id=178-Ugg-Sko-Til-Barn Ugg Sko Til Barn] Øst sa, tar sikte på å utvikle serveringsbransjen i Østen med deltakelse av den østlige provinsen Chief Minister og noen private bedrifter i en miljøvennlige måte Men enkle tiltak som strand opprydding bør . <br><br>Også, glass smykker kan være farlig hvis slått fra treet og . Ærlig talt, det minnet meg om hvordan de tidlige konvensjoner må ha vært med ikke en hel masse store studio deltakelse og noen tilsynelatende tilfeldige boder, ja, det var en messe gjør ekte tatoveringer (Det var alt sterilt og veldig legit, men likevel. <br><br>LinderGanz og GEFEN eksponert rotte baklemmene til trykk størrelsene på 86, 262 og [http://www.clinicafutura.com/images/futura_empresarial/servicos.asp?m=24-Moncler-Jakker-Til-Salg Moncler Jakker Til Salg] 525 mmHg i 2, 4 og 6 timer, henholdsvis [5]. Dette har resultert i at veien dreper [http://www.manausonline.com/prnewswire/colunacapa.asp?ugg=47-Uggs-På-Salg Uggs På Salg] av to leoparder i fjor på BMIC veien (som har brutt korridoren), har påvirket herbivore befolkningen, økte trusler om humanwildlife konflikt, og det er nå en aktiv mulighet for forurensning truer helsen til dyreliv befolkningen ... <br><br>Når du blir vant til å vedlikeholde nettstedet og stadig legge til nytt innhold, ender du opp med å få bruker lojalitet og disse brukerne [http://www.praiabelahotel.com/css/reservas.asp?id=107-Parajumpers-Forhandlere-Norge Parajumpers Forhandlere Norge] kan spre ordet om nettstedet ditt. Når hun hører ordet arbeid, er det ikke gjøre hyggelige assosiasjoner for de fleste menneskelige ånd.<ul>

			<li>[http://220.194.55.213:8086/forum.php?mod=viewthread&tid=2546319 http://220.194.55.213:8086/forum.php?mod=viewthread&tid=2546319]</li>

			<li>[http://observatoiredesreligions.fr/spip.php?article9 http://observatoiredesreligions.fr/spip.php?article9]</li>

			<li>[http://www.xtdwjh.com/service/feedback_look.php?id=468819 http://www.xtdwjh.com/service/feedback_look.php?id=468819]</li>

			<li>[http://www.mariettakaramanli.fr/spip.php?article647/ http://www.mariettakaramanli.fr/spip.php?article647/]</li>

			</ul>

en>LittleWink: WPCleaner v1.31 - Repaired 1 link to disambiguation page - (You can help) - NIF, 2 to be fixed - Conduction, Neutral

2014-03-03T18:06:57Z

WPCleaner v1.31 - Repaired 1 link to disambiguation page - (You can help) - NIF, 2 to be fixed - Conduction, Neutral

Show changes

en>WikiHelper2134: /* Inertial (laser) containment */

2014-02-04T03:17:52Z

Inertial (laser) containment

New page

The '''Baire category theorem''' is an important tool in [[general topology]] and [[functional analysis]]. The theorem has two forms, each of which gives [[sufficient condition]]s for a [[topological space]] to be a [[Baire space]].

The theorem was proved by [[René-Louis Baire]] in his 1899 doctoral thesis.

== Statement of the theorem ==
A [[Baire space]] is a topological space with the following property: for each [[countable]] collection of [[Open set|open]] [[dense set]]s ''Un'', their intersection ∩ ''Un'' is dense.

*('''BCT1''') Every [[complete metric space]] is a Baire space. More generally, every topological space which is [[homeomorphic]] to an [[open set|open subset]] of a [[complete space|complete]] [[pseudometric space]] is a Baire space. Thus every [[completely metrizable]] topological space is a Baire space.
*('''BCT2''') Every [[locally compact]] [[Hausdorff space]] is a Baire space. The proof is similar to the preceding statement; the [[finite intersection property]] takes the role played by completeness.

Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact (the [[irrational number]]s with the metric defined below; also, any [[Banach space]] of infinite dimension), and there are locally compact Hausdorff spaces which are not [[metrizable space|metrizable]] (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in Functional Analysis; the uncountable [[Fort space]]). See [[Counterexamples in Topology|Steen and Seebach]] in the references below.

*('''BCT3''') A non-empty complete metric space is NOT the countable union of [[Nowhere dense set|nowhere-dense]] [[closed set]]s.
This formulation is equivalent to BCT1 and is sometimes more useful in applications. Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has ''non-empty'' interior.

== Relation to the axiom of choice ==
The proofs of '''BCT1''' and '''BCT2''' for arbitrary complete metric spaces require some form of the [[axiom of choice]]; and in fact BCT1 is equivalent over [[Zermelo–Fraenkel set theory|ZF]] to a weak form of the axiom of choice called the [[axiom of dependent choices]].<ref>Blair 1977</ref>

The restricted form of the Baire category theorem in which the complete metric space is also assumed to be separable is provable in ZF with no additional choice principles.<ref>Levy 1979, p. 212</ref> This restricted form applies in particular to the [[real line]], the [[Baire space (set theory)|Baire space]] ωω, and the [[Cantor space]] 2ω.

== Uses of the theorem ==
'''BCT1''' is used in [[functional analysis]] to prove the [[open mapping theorem (functional analysis)|open mapping theorem]], the [[closed graph theorem]] and the [[uniform boundedness principle]].

'''BCT1''' also shows that every complete metric space with no [[isolated point]]s is [[uncountable]]. (If ''X'' is a countable complete metric space with no isolated points, then each [[singleton (mathematics)|singleton]] {''x''} in ''X'' is [[nowhere dense]], and so ''X'' is of [[first category]] in itself.) In particular, this proves that the set of all [[real number]]s is uncountable.

'''BCT1''' shows that each of the following is a Baire space:
* The space '''R''' of [[real number]]s
* The [[irrational number]]s, with the metric defined by ''d''(''x'', ''y'') = 1 / (''n'' + 1), where ''n'' is the first index for which the [[continued fraction]] expansions of ''x'' and ''y'' differ (this is a complete metric space)
* The [[Cantor set]]

By '''BCT2''', every finite-dimensional Hausdorff [[manifold]] is a Baire space, since it is locally compact and Hausdorff. This is so even for non-[[paracompact]] (hence nonmetrizable) manifolds such as the [[long line (topology)|long line]].

== Proof ==
The following is a standard proof that a complete pseudometric space <math>\scriptstyle X</math> is a Baire space.

Let <math>\scriptstyle U_n</math> be a countable collection of open dense subsets. We want to show that the intersection <math>\scriptstyle \bigcap U_n</math> is dense. A subset is dense if and only if every nonempty open subset intersects it. Thus, to show that the intersection is dense, it is sufficient to show that any nonempty open set <math>\scriptstyle W</math> has a point <math>\scriptstyle x</math> in common with all of the <math>\scriptstyle U_n</math>. Since <math>\scriptstyle U_1</math> is dense, <math>\scriptstyle W</math> intersects <math>\scriptstyle U_1</math>; thus, there is a point <math>\scriptstyle x_1</math> and <math>\scriptstyle 0 \;<\; r_1 \;<\; 1</math> such that:
:<math>\overline{B}(x_1, r_1) \subset W \cap U_1</math>

where <math>\scriptstyle B(x, r)</math> and <math>\scriptstyle \overline{B}(x, r)</math> denote an open and closed ball, respectively, centered at <math>\scriptstyle x</math> with radius <math>\scriptstyle r</math>. Since each <math>\scriptstyle U_n</math> is dense, we can continue recursively to find a pair of sequences <math>\scriptstyle x_n</math> and <math>\scriptstyle 0 \;<\; r_n \;<\; \frac{1}{n}</math> such that:
:<math>\overline{B}(x_n, r_n) \subset B(x_{n - 1}, r_{n - 1}) \cap U_n</math>

(This step relies on the axiom of choice.) Since <math>\scriptstyle x_n \;\in\; B(x_m, r_m)</math> when <math>\scriptstyle n \;>\; m</math>, we have that <math>\scriptstyle x_n</math> is [[Cauchy sequence|Cauchy]], and hence <math>\scriptstyle x_n</math> converges to some limit <math>x</math> by completeness. For any <math>\scriptstyle n</math>, by closedness,
:<math>x \in \overline{B}(x_n, r_n).</math>

Therefore <math>\scriptstyle x \;\in\; W</math> and <math>\scriptstyle x \;\in\; U_n</math> for all <math>\scriptstyle n</math>.

== See also ==
* [[Property of Baire]]

== Notes ==
{{reflist}}

== References ==
*R. Baire. [http://books.google.com/books?id=cS4LAAAAYAAJ Sur les fonctions de variables réelles.] Ann. di Mat., 3:1–123, 1899.
* Blair, Charles E. (1977), "The Baire category theorem implies the principle of dependent choices.", ''Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.'', v. 25 n. 10, pp. 933–934.
* [[Azriel Levy|Levy, Azriel]] (1979), ''Basic Set Theory''. Reprinted by Dover, 2002. ISBN 0-486-42079-5
* [[Eric Schechter|Schechter, Eric]], ''Handbook of Analysis and its Foundations'', Academic Press, ISBN 0-12-622760-8
* [[Lynn Steen|Lynn Arthur Steen]] and [[J. Arthur Seebach, Jr.]], ''[[Counterexamples in Topology]]'', Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).

==External links==
*T. Tao, [http://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/ 245B, Notes 9: The Baire category theorem and its Banach space consequences]
* [http://www.encyclopediaofmath.org/index.php/Baire_theorem Encyclopaedia of Mathematics article on Baire theorem]

{{DEFAULTSORT:Baire Category Theorem}}
[[Category:General topology]]
[[Category:Functional analysis]]
[[Category:Theorems in topology]]
[[Category:Articles containing proofs]]