Transfinite induction: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Blaisorblade
→‎Transfinite recursion: Mark as {{technical}}
No edit summary
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{Separation axiom}}
== ZbkTLk Jordan 6 World Cup Brazil Pog1NBqj ==
In [[topology]] and related branches of [[mathematics]], '''Tychonoff spaces''' and '''completely regular spaces''' are kinds of [[topological space]]s.
These conditions are examples of [[separation axiom]]s.


Tychonoff spaces are named after [[Andrey Nikolayevich Tychonoff]], whose [[Russian language|Russian]] name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov" etc.
If you say yes, you are a "moralist" in the manner of President George W. Bush, who saw things in black and white, called Saddam Hussein evil, and invaded Iraq and Afghanistan with the stated goal of uprooting millennia of habits and installing American style democracy. power  and the world's demand for it.<br><br>How can we and what will we make it? The challenges we set for ourselves, the goals that we establish, the things that we accomplish will be determined by the effort that we make and the opportunities that are provided to us. So this is an opportunity to start again and look at a variety of different things that I am going to talk about a little bit later: The academic planning team, a group that has been working throughout the summer to help us develop an academic plan, built upon what we have done in the past with an eye on the future and what we might become as an institution. <br><br>Forward Davante Gardner, who scored 26 points when these teams met back in February, leads Marquette with nine points and four rebounds. James Southerland has a team high eight points for Syracuse, which entered halftime with a 24 18 lead despite allowing 10 offensive rebounds. Marquertte shot just 27 percent, 1 of 9 from three point range in the first half.<br><br>Venturi first gained national attention in 1956 when, as an amateur, he finished second in that year's Masters after .The clubs incorporate everything that Venturi venturia tube with a decrease in the inside diameter that is used to increase the flow velocity of the fluid and thereby cause a pressure drop; used to measure the flow velocity (a venturimeter) or to draw another fluid into the stream. <br><br>The second oldest hotel in [http://www.ibewlocal58.org/answerss.php??p=59042-buy-authentic-outlet-Nike-Shox-R4-alry-White-Yellow-online-2014.html buy authentic outlet Nike Shox R4 alry White Yellow online 2014] the United States, the Equinox has enjoyed a rich history that adds to its historic [http://www.ibewlocal7.com/themess.php??p=37680-cheap-authentic-Nike-Air-Max-90-White-Blue-Black-running-sneaker.html cheap authentic Nike Air Max 90 White Blue Black running sneaker] appeal. Presidents William Howard Taft, Ulysses S. Grant, Theodore Roosevelt and Benjamin Harrison were all guests over the years. Abraham Lincoln was almost one, the story goes. His wife, Mary Todd Lincoln, vacationed with her two sons in 1864 and [http://www.ibewlocal58.org/answerss.php??p=30710-Website-To-Pre-Order-new-Nike-Womens-Dunk-High-Skinny-Supreme-Orange-Black-Peach-.html Website To Pre Order new Nike Womens Dunk High Skinny Supreme Orange Black Peach] was so enamored of the place that she made plans to come back with her husband.<br><br>The visit to Gore Island is expected to be a particularly significant part of the trip for Obama and his family as it marks an important site for both Africans and African Americans as the home of the House of Slaves, where men, women and children were kept in small cells before being shipped off to the Americas. Presidents Bill Clinton and George W. Bush, as well as former South African President Nelson Mandela.<br><br>Robbins Elementary School in Prichard presented its second annual Popcorn Parade recently. Kindergarten students were asked to learn 25 "popcorn" or sight words. Those who could recite all the words got to participate in the parade. Students from all other grade levels lined up around the campus to receive the goodies thrown by the kindergartners. Students sold pumpkins and observed "Jeans for Babies" Day to raise the funds they presented to Jennifer Bitner, Southwest Division Director for March of Dimes. Chase Butler is FBLA Satsuma Chapter president.
相关的主题文章:
<ul>
 
  <li>[http://www.0772sn.com/thread-192800-1-1.html http://www.0772sn.com/thread-192800-1-1.html]</li>
 
  <li>[http://hhsysw.com/shownews.asp?id=68 http://hhsysw.com/shownews.asp?id=68]</li>
 
  <li>[http://thebuddypalace.com/groups/bez5ni-jordan-6-brazil-ivzui7jw/ http://thebuddypalace.com/groups/bez5ni-jordan-6-brazil-ivzui7jw/]</li>
 
</ul>


== Definitions ==
== PgKB8p Jordan 6 Brazil V55VklWG ==


Suppose that ''X'' is a topological space.
Message to union leadership is I would very much encourage them not to ask their members to undertake illegal strike activity. OSSTF and the ETFO have already voted in favour of a one day protest in the new year. The unions have also vowed to extend their boycott of extracurricular activities.<br><br>Jon Krawczynski's AP story on the U of M's hiring Rick Pitino's son as its new basketball coach says: "Richard Pitino was regarded as one of the nation's up and coming young coaches at Florida International and comes with a last name that draws immediate respect in all corners of college basketball. That combination was too good for Minnesota to pass up. . Many expected Teague and associate athletic director Mike Ellis  two men considered to have strong contacts in the college basketball coaching world  to go for a big name to bring a different energy into a program they felt had hit a wall."<br><br>As President Bush himself has said, "America must and will keep its word to the men and women who have given us so much. So even as we Americans may at times disagree on matters of foreign policy, we share a profound respect and reverence for the men and women of our military and their families. <br><br>SUPER MARKET 40WEST FOURTH ST. MANSRELD IT'S ALWAYS FRESHER AT BUEHLER'S Xevr Eastern Shore POTATOES 10 23c Jumbo Vine Ripened CANTALOUPES 2 23c Watermelons I L. M. Laughbauni. 161 Sturges avenue, is week ending in Chica  go. III. Adario BIUEFS Mrs. Ralph Kyle ?nd Mrs. Mat  tic Hoffman of Mansfield were callers at the R. <br><br>You ask any guy growing up what the worst insult was and it's always, "You throw like a girl."I know I've heard that plenty of times, but I'd never really thought about it like that.You look at every horrific act [http://www.ibewlocal529.ca/indexss.php??p=2372-Nike-Air-Max-95-Black-Black-mens-shoes-on-sale.html Nike Air Max 95 Black Black mens shoes on sale] and they are never perpetrated by a woman. The Boston Marathon bombings? You didn't even consider that it was a woman. <br><br>That last bit may be wishful thinking. Las Vegas gaming revenue for 2009, was down more than 10 percent through November, though there were signs of recovery late in the year. Whether or not the president was to blame, local resident Christie Terronial(ph) says Mr. Obama really needs to watch what he says.<br><br>I really love the fact that issues like these are hot button issues that ignorant racists can't ignore and it forces them to expose their [http://www.ibewlocal58.org/answerss.php??p=7027-Nike-Free-Run-2-2012-Royal-Blue-Navy-White-size-6.html Nike Free Run 2 2012 Royal Blue Navy White size 6] bias in public. People who believe that white people have gotten beyond the issue of race only need to read what you people are writing. sad.<br><br>But Obama has also argued that Washington's 51 year economic embargo on Cuba should remain in force, and his administration has imposed tens of millions of dollars in fines on international companies for [http://www.ibewlocal7.com/themess.php??p=55020-buy-cheap-New-Nike-Air-Max-90-White-Black-Grey-Yellow-Release-Date-2014-Online.html buy cheap New Nike Air Max 90 White Black Grey Yellow Release Date 2014 Online] violating the sanctions. government development subcontractor Alan Gross put relations back in a deep freeze. Gross remains jailed, but this year Washington decided it would no longer let the case stand in the way on areas of common interest. and Cuba have held multiple rounds of talks on restoring direct mail service and immigration issues, with more scheduled for January. Diplomats on both sides report cordial relations and call each other at home. The two nations' coast guards reportedly work well together on things like drug interdiction.
 
相关的主题文章:
''X'' is a ''completely regular space'' if given any [[closed set]] ''F'' and any [[Point (geometry)|point]] ''x'' that does not belong to ''F'', then [[existential quantification|there is]] a [[continuous function (topology)|continuous function]] ''f'' from ''X'' to the [[real line]] '''R''' such that ''f''(''x'') is {{num|0}} and, [[universal quantification|for every]] ''y'' in ''F'', ''f''(''y'') is {{num|1}}.
<ul>
In other terms, this condition says that ''x'' and ''F'' can be [[separated sets|separated by a continuous function]].
 
 
  <li>[http://www.datasheetwiki.com/index.php/User:Bcdlrbdaow#Yx0hb4_World_Cup_Brazil_6s_sbaJV4UY http://www.datasheetwiki.com/index.php/User:Bcdlrbdaow#Yx0hb4_World_Cup_Brazil_6s_sbaJV4UY]</li>
''X'' is a ''Tychonoff space'', or ''T<sub>3½</sub> space'', or ''T<sub>π</sub> space'', or ''completely T<sub>3</sub> space'' if it is both completely regular and [[Hausdorff space|Hausdorff]].
 
 
  <li>[http://www.wakeariwonderland.com/wiki/index.php?title=User:Bghcuurzzw#kxogjE_Jordan_6_World_Cup_Brazil_g2xOQRgG http://www.wakeariwonderland.com/wiki/index.php?title=User:Bghcuurzzw#kxogjE_Jordan_6_World_Cup_Brazil_g2xOQRgG]</li>
Note that some mathematical literature uses different definitions for the term "completely regular" and the terms involving "T".
 
The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two kinds of terms, or use all terms synonymously for only one condition.
  <li>[http://www.xiaoyuankeji8.com/shownews.asp?id=90 http://www.xiaoyuankeji8.com/shownews.asp?id=90]</li>
In Wikipedia, we will use the terms "completely regular" and "Tychonoff" freely, but we'll avoid the less clear "T" terms.
 
In other literature, you should take care to find out which definitions the author is using.
</ul>
(The phrase "completely regular Hausdorff", however, is unambiguous, and always means a Tychonoff space.)
For more on this issue, see [[History of the separation axioms]].
 
Completely regular spaces and Tychonoff spaces are related through the notion of [[Kolmogorov equivalence]].
A topological space is Tychonoff if and only if it's both completely regular and [[Kolmogorov space|T<sub>0</sub>]].
On the other hand, a space is completely regular if and only if its [[Kolmogorov quotient]] is Tychonoff.
 
== Examples and counterexamples ==
 
Almost every topological space studied in [[mathematical analysis]] is Tychonoff, or at least completely regular.
For example, the [[real line]] is Tychonoff under the standard [[Euclidean space|Euclidean topology]].
Other examples include:
 
* Every [[metric space]] is Tychonoff; every [[pseudometric space]] is completely regular.
* Every [[locally compact]] [[regular space|regular]] space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
* In particular, every [[topological manifold]] is Tychonoff.
* Every [[totally ordered set]] with the [[order topology]] is Tychonoff.
* Every [[topological group]] is completely regular.
* Generalising both the metric spaces and the topological groups, every [[uniform space]] is completely regular. The converse is also true: every completely regular space is uniformisable.
* Every [[CW complex]] is Tychonoff.
* Every [[normal space|normal]] regular space is completely regular, and every normal Hausdorff space is Tychonoff.
* The [[Niemytzki plane]] is an example of a Tychonoff space which is not [[normal space|normal]].
 
== Properties ==
 
===Preservation===
 
Completely regularity and the Tychonoff property are well-behaved with respect to [[initial topology|initial topologies]]. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies. It follows that:
* Every [[subspace (topology)|subspace]] of a completely regular or Tychonoff space has the same property.
* A nonempty [[product space]] is completely regular (resp. Tychonoff) if and only if each factor space is completely regular (resp. Tychonoff).
 
Like all separation axioms, completely regularity is not preserved by taking [[final topology|final topologies]]. In particular, [[Quotient space|quotients]] of completely regular spaces need not be [[regular space|regular]]. Quotients of Tychonoff spaces need not even be [[Hausdorff space|Hausdorff]]. There are closed quotients of the [[Moore plane]] which provide counterexamples.
 
===Real-valued continuous functions===
 
For any topological space ''X'', let ''C''(''X'') denote the family of real-valued [[continuous function (topology)|continuous function]]s on ''X'' and let ''C''*(''X'') be the subset of [[bounded function|bounded]] real-valued continuous functions.
 
Completely regular spaces can be characterized by the fact that their topology is completely determined by ''C''(''X'') or ''C''*(''X''). In particular:
 
*A space ''X'' is completely regular if and only if it has the [[initial topology]] induced by ''C''(''X'') or ''C''*(''X'').
*A space ''X'' is completely regular if and only if every closed set can be written as the intersection of a family of [[zero set]]s in ''X'' (i.e. the zero sets form a basis for the closed sets of ''X'').
*A space ''X'' is completely regular if and only if the [[cozero set]]s of ''X'' form a [[basis (topology)|basis]] for the topology of ''X''.
 
Given an arbitrary topological space (''X'', τ) there is a universal way of associating a completely regular space with (''X'', τ). Let ρ be the initial topology on ''X'' induced by ''C''<sub>τ</sub>(''X'') or, equivalently, the topology generated by the basis of cozero sets in (''X'', τ). Then ρ will be the [[finest topology|finest]] completely regular topology on ''X'' which is coarser than τ. This construction is [[universal property|universal]] in the sense that any continuous function
:<math>f:(X,\tau)\to Y</math>
to a completely regular space ''Y'' will be continuous on (''X'', ρ). In the language of [[category theory]], the [[functor]] which sends (''X'', τ) to (''X'', ρ) is [[left adjoint]] to the inclusion functor '''CReg''' &rarr; '''Top'''. Thus the category of completely regular spaces '''CReg''' is a [[reflective subcategory]] of '''Top''', the [[category of topological spaces]]. By taking [[Kolmogorov quotient]]s, one sees that the subcategory of Tychonoff spaces is also reflective.
 
One can show that ''C''<sub>τ</sub>(''X'') = ''C''<sub>ρ</sub>(''X'') in the above construction so that the rings ''C''(''X'') and ''C''*(''X'') are typically only studied for completely regular spaces ''X''.
 
The category of [[Realcompact_space|real compact]] Tychonoff spaces is anti-equivalent to the category of the rings ''C''(''X'') (where ''X'' is real compact) together with ring homomorphisms as maps. For example one can reconstruct $X$ from ''C''(''X'') when ''X'' is (real) compact. The algebraic theory of these rings is therefore subject of intensive studies.  
A vast generalisation of this class of rings which still resembles many properties of Tychonoff spaces but is also applicable in [[real algebraic geometry]], is the class of [[real closed ring]]s.
 
===Embeddings===
 
Tychonoff spaces are precisely those spaces which can be
[[embedding|embedded]] in [[compact Hausdorff space]]s. More precisely, for every Tychonoff space ''X'', there exists a compact Hausdorff space ''K'' such that ''X'' is [[homeomorphic]] to a subspace of ''K''.
 
In fact, one can always choose ''K'' to be a [[Tychonoff cube]] (i.e. a possibly infinite product of [[unit interval]]s). Every Tychonoff cube is compact Hausdorff as a consequence of [[Tychonoff's theorem]]. Since every subspace of a compact Hausdorff space is Tychonoff one has:
 
:''A topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube''.
 
===Compactifications===
 
Of particular interest are those embeddings where the image of ''X'' is [[dense subset|dense]] in ''K''; these are called Hausdorff [[Compactification (mathematics)|compactifications]] of ''X''. Given any embedding of a Tychonoff space ''X'' in a compact Hausdorff space ''K'' the [[closure (topology)|closure]] of the image of ''X'' in ''K'' is a compactification of ''X''.
 
Among those Hausdorff compactifications, there is a unique "most general" one, the [[Stone–Čech compactification]] β''X''.
It is characterised by the [[universal property]] that, given a continuous map ''f'' from ''X'' to any other compact Hausdorff space ''Y'', there is a [[unique]] continuous map ''g'' from β''X'' to ''Y'' that extends ''f'' in the sense that ''f'' is the [[composition (functions)|composition]] of ''g'' and ''j''.
 
===Uniform structures===
 
Complete regularity is exactly the condition necessary for the existence of [[uniform structure]]s on a topological space. In other words, every [[uniform space]] has a completely regular topology and every completely regular space ''X'' is [[uniformizable]]. A topological space admits a separated uniform structure if and only if it is Tychonoff.
 
Given a completely regular space ''X'' there is usually more than one uniformity on ''X'' that is compatible with the topology of ''X''. However, there will always be a finest compatible uniformity, called the [[fine uniformity]] on ''X''. If ''X'' is Tychonoff, then the uniform structure can be chosen so that β''X'' becomes the [[completion (topology)|completion]] of the uniform space ''X''.
 
==References==
* Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
* [[Leonard Gillman|Gillman, Leonard]]; [[Meyer Jerison|Jerison, Meyer]] Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp
 
[[Category:Topology]]
[[Category:Separation axioms]]
[[Category:Topological spaces]]

Latest revision as of 00:27, 24 October 2014

ZbkTLk Jordan 6 World Cup Brazil Pog1NBqj

If you say yes, you are a "moralist" in the manner of President George W. Bush, who saw things in black and white, called Saddam Hussein evil, and invaded Iraq and Afghanistan with the stated goal of uprooting millennia of habits and installing American style democracy. power and the world's demand for it.

How can we and what will we make it? The challenges we set for ourselves, the goals that we establish, the things that we accomplish will be determined by the effort that we make and the opportunities that are provided to us. So this is an opportunity to start again and look at a variety of different things that I am going to talk about a little bit later: The academic planning team, a group that has been working throughout the summer to help us develop an academic plan, built upon what we have done in the past with an eye on the future and what we might become as an institution.

Forward Davante Gardner, who scored 26 points when these teams met back in February, leads Marquette with nine points and four rebounds. James Southerland has a team high eight points for Syracuse, which entered halftime with a 24 18 lead despite allowing 10 offensive rebounds. Marquertte shot just 27 percent, 1 of 9 from three point range in the first half.

Venturi first gained national attention in 1956 when, as an amateur, he finished second in that year's Masters after .The clubs incorporate everything that Venturi venturia tube with a decrease in the inside diameter that is used to increase the flow velocity of the fluid and thereby cause a pressure drop; used to measure the flow velocity (a venturimeter) or to draw another fluid into the stream.

The second oldest hotel in buy authentic outlet Nike Shox R4 alry White Yellow online 2014 the United States, the Equinox has enjoyed a rich history that adds to its historic cheap authentic Nike Air Max 90 White Blue Black running sneaker appeal. Presidents William Howard Taft, Ulysses S. Grant, Theodore Roosevelt and Benjamin Harrison were all guests over the years. Abraham Lincoln was almost one, the story goes. His wife, Mary Todd Lincoln, vacationed with her two sons in 1864 and Website To Pre Order new Nike Womens Dunk High Skinny Supreme Orange Black Peach was so enamored of the place that she made plans to come back with her husband.

The visit to Gore Island is expected to be a particularly significant part of the trip for Obama and his family as it marks an important site for both Africans and African Americans as the home of the House of Slaves, where men, women and children were kept in small cells before being shipped off to the Americas. Presidents Bill Clinton and George W. Bush, as well as former South African President Nelson Mandela.

Robbins Elementary School in Prichard presented its second annual Popcorn Parade recently. Kindergarten students were asked to learn 25 "popcorn" or sight words. Those who could recite all the words got to participate in the parade. Students from all other grade levels lined up around the campus to receive the goodies thrown by the kindergartners. Students sold pumpkins and observed "Jeans for Babies" Day to raise the funds they presented to Jennifer Bitner, Southwest Division Director for March of Dimes. Chase Butler is FBLA Satsuma Chapter president. 相关的主题文章:

PgKB8p Jordan 6 Brazil V55VklWG

Message to union leadership is I would very much encourage them not to ask their members to undertake illegal strike activity. OSSTF and the ETFO have already voted in favour of a one day protest in the new year. The unions have also vowed to extend their boycott of extracurricular activities.

Jon Krawczynski's AP story on the U of M's hiring Rick Pitino's son as its new basketball coach says: "Richard Pitino was regarded as one of the nation's up and coming young coaches at Florida International and comes with a last name that draws immediate respect in all corners of college basketball. That combination was too good for Minnesota to pass up. . Many expected Teague and associate athletic director Mike Ellis two men considered to have strong contacts in the college basketball coaching world to go for a big name to bring a different energy into a program they felt had hit a wall."

As President Bush himself has said, "America must and will keep its word to the men and women who have given us so much. So even as we Americans may at times disagree on matters of foreign policy, we share a profound respect and reverence for the men and women of our military and their families.

SUPER MARKET 40WEST FOURTH ST. MANSRELD IT'S ALWAYS FRESHER AT BUEHLER'S Xevr Eastern Shore POTATOES 10 23c Jumbo Vine Ripened CANTALOUPES 2 23c Watermelons I L. M. Laughbauni. 161 Sturges avenue, is week ending in Chica go. III. Adario BIUEFS Mrs. Ralph Kyle ?nd Mrs. Mat tic Hoffman of Mansfield were callers at the R.

You ask any guy growing up what the worst insult was and it's always, "You throw like a girl."I know I've heard that plenty of times, but I'd never really thought about it like that.You look at every horrific act Nike Air Max 95 Black Black mens shoes on sale and they are never perpetrated by a woman. The Boston Marathon bombings? You didn't even consider that it was a woman.

That last bit may be wishful thinking. Las Vegas gaming revenue for 2009, was down more than 10 percent through November, though there were signs of recovery late in the year. Whether or not the president was to blame, local resident Christie Terronial(ph) says Mr. Obama really needs to watch what he says.

I really love the fact that issues like these are hot button issues that ignorant racists can't ignore and it forces them to expose their Nike Free Run 2 2012 Royal Blue Navy White size 6 bias in public. People who believe that white people have gotten beyond the issue of race only need to read what you people are writing. sad.

But Obama has also argued that Washington's 51 year economic embargo on Cuba should remain in force, and his administration has imposed tens of millions of dollars in fines on international companies for buy cheap New Nike Air Max 90 White Black Grey Yellow Release Date 2014 Online violating the sanctions. government development subcontractor Alan Gross put relations back in a deep freeze. Gross remains jailed, but this year Washington decided it would no longer let the case stand in the way on areas of common interest. and Cuba have held multiple rounds of talks on restoring direct mail service and immigration issues, with more scheduled for January. Diplomats on both sides report cordial relations and call each other at home. The two nations' coast guards reportedly work well together on things like drug interdiction. 相关的主题文章:

Retrieved from "https://en.formulasearchengine.com/index.php?title=Transfinite_induction&oldid=221199"