Homeomorphism: Difference between revisions

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In [[mathematics]], the '''Hausdorff maximal principle''' is an alternate and earlier formulation of [[Zorn's lemma]] proved by [[Felix Hausdorff]] in 1914 (Moore 1982:168). It states that in any [[partial order|partially ordered set]], every [[total order|totally ordered]] [[subset]] is contained in a maximal totally ordered subset.
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The Hausdorff maximal principle is one of many statements equivalent to the [[axiom of choice]] over [[Zermelo–Fraenkel set theory]]. The principle is also called the '''Hausdorff maximality theorem''' or the '''Kuratowski lemma''' (Kelley 1955:33).
 
==Statement==
 
The Hausdorff maximal principle states that, in any [[partial order|partially ordered set]], every [[total order|totally ordered]] [[subset]] is contained in a maximal totally ordered subset. Here a maximal totally-ordered subset is one that, if enlarged in any way, does not remain totally ordered. The maximal set produced by the principle is not unique, in general; there may be many maximal totally ordered subsets containing a given totally ordered subset.
 
An equivalent form of the principle is that in every partially ordered set there exists a maximal totally ordered subset.
 
To prove that it follows from the original form, let ''A'' be a [[poset]]. Then <math>\varnothing</math> is a totally ordered subset of ''A'', hence there exists a maximal totally ordered subset containing <math>\varnothing</math>, in particular ''A'' contains a maximal totally ordered subset.
 
For the converse direction, let ''A'' be a partially ordered set and ''T'' a totally ordered subset of ''A''. Then
:<math>\{S\mid T\subseteq S\subseteq A\mbox{ and S totally ordered}\}</math>
is partially ordered by set inclusion <math>\subseteq</math>, therefore it contains a maximal totally ordered subset ''P''. Then the set <math>M=\bigcup P</math> satisfies the desired properties.
 
The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is very similar to this proof.
 
==References==
* John Kelley (1955), ''General topology'', Von Nostrand.  
* Gregory Moore (1982), ''Zermelo's axiom of choice'', Springer.
 
==External links==
* {{planetmath reference|id=3491|title=Hausdorff's maximum principle}}
* {{planetmath reference|id=3493|title=A proof of equivalence of Zorn's lemma, the well-ordering theorem, and Hausdorff's maximum principle}}
 
[[Category:Axiom of choice]]
[[Category:Order theory]]
[[Category:Mathematical principles]]

Revision as of 04:22, 6 February 2014

Msvcr71.dll is an important file that helps support Windows process different components of the program including important files. Specifically, the file is used to help run corresponding files in the "Virtual C Runtime Library". These files are significant in accessing any settings which support the different applications and programs inside the system. The msvcr71.dll file fulfills many significant functions; but it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer might have a hard time processing plus reading components of the system. Nonetheless, users want not panic considering this problem may be solved by following several procedures. And I might show we several tricks regarding Msvcr71.dll.

Registry is not moreover significant to fast computer boot up, however also crucial to the performance of a computer. If you have a registry error, you could face blue screen, freezing or even crash. It's necessary to frequently clean up the invalid, missing, junk registry keys to keep the computer healthy and running quick.

Perfect Optimizer moreover has to remove junk files plus is completely Windows Vista compatible. Most registry product just don't have the time plus income to research Windows Vista errors. Because best optimizer has a large customer base, they do have the time, funds and reasons to help completely support Windows Vista.

The problem with most of the folks is that they never desire to spend cash. In the cracked adaptation 1 does not have to pay anything and may download it from internet fairly easily. It is simple to install as well. But, the problem comes whenever it is very unable to detect all possible viruses, spyware plus malware in the system. This is considering it's obsolete in nature and does not receive any usual updates within the website downloaded. Thus, your program is accessible to problems like hacking.

After which, I furthermore bought the Regtool zookaware Software, and it further secure my computer having program crashes. All my registry problems are fixed, and I can work peacefully.

The software finds these difficulties and takes care of them in the shortest time possible. The hard drive may also result issues sometimes, specifically should you have 1 that is virtually maxed. When a start a machine, the are numerous booting processes included plus having an virtually full storage refuses to assist a bit. You will always have a slow PC because there are a lot of aspects in the hard disk being processed simultaneously. The best method to solve this problem is to upgrade. This allows a PC several time to breath plus functioning quicker instantly.

Maybe you're asking why these windows XP error messages appear. Well, for you to be capable to know the fix, you must initially understand where those mistakes come from. There is this software called registry. A registry is software which shops everything on a PC from a usual configuration, setting, info, and logs of activities from installing to UN-installing, saving to deleting, and a lot more alterations you do inside the program pass from it plus gets 'tagged' plus saved because a simple file for recovery purposes. Imagine it because a big recorder, a registrar, of all the records inside a PC.

Most persons create the mistake of striving to fix Windows registry by hand. I strongly recommend we don't do it. Unless you are a computer expert, I bet you'll spend hours plus hours learning the registry itself, let alone fixing it. And why if you waste the valuable time in understanding plus fixing anything you recognize nothing about? Why not let a smart plus specialist registry cleaner do it for you? These software programs could do the job inside a far better method! Registry cleaners are very affordable as well; you pay a once fee and utilize it forever. Additionally, most specialist registry products are pretty reliable and effortless to use. If you require more information on how to fix Windows registry, merely visit my website by clicking the link below!