Brahmagupta–Fibonacci identity: Difference between revisions

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{{Separation axiom}}
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In [[topology]], an '''Urysohn space''', or '''T<sub>2½</sub> space''', is a [[topological space]] in which any two distinct points can be [[separated by closed neighborhoods]]. A '''completely Hausdorff space''', or '''functionally Hausdorff space''', is a topological space in which any two distinct points can be separated by a [[continuous function]]. These conditions are [[separation axiom]]s that are somewhat stronger than the more familiar [[Hausdorff space|Hausdorff axiom]] T<sub>2</sub>.
 
==Definitions==
 
Suppose that ''X'' is a [[topological space]]. Let ''x'' and ''y'' be points in ''X''.
*We say that ''x'' and ''y'' can be ''[[separated by closed neighborhoods]]'' if there exists a [[closed set|closed]] [[neighborhood (topology)|neighborhood]] ''U'' of ''x'' and a closed neighborhood ''V'' of ''y'' such that ''U'' and ''V'' are [[disjoint sets|disjoint]] (''U'' ∩ ''V'' = ∅). (Note that a "closed neighborhood of ''x''" is a [[closed set]] that contains an [[open set]] containing ''x''.)
*We say that ''x'' and ''y'' can be ''[[separated by a function]]'' if there exists a [[continuity (topology)|continuous function]] ''f'' : ''X'' → [0,1] (the [[unit interval]]) with ''f''(''x'') = 0 and ''f''(''y'') = 1.
 
A '''Urysohn space''', or '''T<sub>2½</sub> space''', is a space in which any two distinct points can be separated by closed neighborhoods.  
 
A '''completely Hausdorff space''', or '''functionally Hausdorff space''', is a space in which any two distinct points can be separated by a continuous function.
 
==Naming conventions==
 
The study of separation axioms is notorious for conflicts with naming conventions used. The definitions used in this article are those given by Willard (1970) and are the more modern definitions. Steen and Seebach (1970) and various other authors reverse the definition of completely Hausdorff spaces and Urysohn spaces. Readers of textbooks in topology must be sure to check the definitions used by the author. See [[History of the separation axioms]] for more on this issue.
 
==Relation to other separation axioms==
 
It is an easy exercise to show that any two points which can be separated by a function can be separated by closed neighborhoods. If they can be separated by closed neighborhoods then clearly they can be separated by neighborhoods. It follows that every completely Hausdorff space is Urysohn and every Urysohn space is [[Hausdorff space|Hausdorff]].
 
One can also show that every [[regular Hausdorff space]] is Urysohn and every [[Tychonoff space]] (=completely regular Hausdorff space) is completely Hausdorff. In summary we have the following implications:
<center>
{| style="text-align: center;"
|-
| [[Tychonoff space|Tychonoff]] (T<sub>3½</sub>) || &nbsp;<math>\Rightarrow</math>&nbsp;
| [[regular space|regular Hausdorff]] (T<sub>3</sub>)
|-
|<math>\Downarrow</math> || || <math>\Downarrow</math>
|-
| completely Hausdorff || &nbsp;<math>\Rightarrow</math>&nbsp;
| Urysohn (T<sub>2½</sub>) || &nbsp;<math>\Rightarrow</math>&nbsp;
| [[Hausdorff space|Hausdorff]] (T<sub>2</sub>) || &nbsp;<math>\Rightarrow</math>&nbsp;
| [[T1 space|T<sub>1</sub>]]
|}
</center>
One can find counterexamples showing that none of these implications reverse.<ref>{{planetmath reference|id=5718|title=Hausdorff space not completely Hausdorff}}</ref>
 
==Examples==
 
The [[cocountable extension topology]] is the topology on the [[real line]] generated by the [[union (set theory)|union]] of the usual [[Euclidean topology]] and the [[cocountable topology]]. Sets are [[open set|open]] in this topology if and only if they are of the form ''U'' \ ''A'' where ''U'' is open in the Euclidean topology and ''A'' is [[countable]]. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff).
 
There are obscure examples of spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff. For details see Steen and Seebach.
 
==Notes==
<references/>
 
==References==
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
*Stephen Willard, ''General Topology'', Addison-Wesley, 1970. Reprinted by Dover Publications, New York, 2004. ISBN 0-486-43479-6 (Dover edition).
 
*{{planetmath reference|id=5717|title=Completely Hausdorff}}
 
[[Category:Separation axioms]]

Latest revision as of 22:24, 26 October 2014

There are numerous articles on fat loss that have been composed. We have read a number of them which suggest a diet for those wishing to get rid of fat. Others have spoken of exercise. We have several who recommend you hiring a individual trainer. The query, though, nevertheless remains. What is healthy fat reduction? What is healthy weight for ladies?

Males have more muscle mass than females. This fact is based on hormones each body makes. Female bodies provide estrogen and guy bodies produce testosterone. Estrogen is responsible for women storing body fat in their hips plus chest. This strategic location of fat makes a woman capable of carrying and feeding a baby. Testosterone assists build strong muscles inside the arms and chest. Historically, guys have been classified because hunters plus gatherers. They are the physically stronger sex plus their genetic makeup is ideal for that character. So, does using the same gender neutral bmi chart for both guys and women create sense? No.

37. Making time for yourself: Running = sanity. Alone or with neighbors it has great therapeutic results which last all day bmi chart men. I find doing it early in the morning is best because I recognize I'll receive my run inside and "lifetime stuff" during the day will likely not get inside the means.

Naturally, its not always black plus white. We recognize which females tend to have a higher percentage of body fat than guys, even if they are the same weight and height, and therefore have the same BMI. (Sorry, ladies.) Also, the elder we receive, you tend to have more body fat than younger adults. And again, athletes plus extremely muscular individuals like bodybuilders will likely not have an accurate BMI according to the way.

Anorexia nervosa is an emotional disorder where the leading focus can be on food / the avoidance of food however it also deals with harmful ways of gaining perfection plus a desire to control items. In a society which associates unreasonable thinness with beauty, there has been a marked escalation in the number of young adolescent ladies with anorexia. Many of them die due to starvation-related causes, suffer from bodily complications, or end up committing suicide. It is important to treat such individuals with psychotherapy, family therapy, plus medication.

There is equally a certain amount of excitement that comes along with all of the secrecy associated bmi chart women. The two of we might meet up with each alternative for a fast rendezvous inside a hotel space or you might go someplace secluded for a weekend away together. It virtually doesnt matter where you meet up with every alternative. Just the truth which you are getting away with anything may be exciting for the both of you.

Let's break it down a bit. "Ideal body weight" is defined as the weight persons are expected to weigh for wise wellness based on age, sex plus height. Body composition is the makeup of the body in terms of fat-free mass (the piece of the body composition which represents everything however, fat--blood, bones, connective cells, organs plus muscles) and fat mass (actual amount of essential plus non-essential fat in the body.) Essential fat is the amount of fat mandatory for usual functioning of the body.

In addition to weight training and aerobic exercise, overcoming a taste for fat may help you slim down. Your ideal women's clothing size might depend on whether we are tiny, medium or big frame; whether you're a pear or hourglass form plus whether you have excess skin/fat from pregnancies or being fat. Maintain wise exercise plus eating behavior and we might find a fresh skinny jean size which proves it's not all inside the genes!

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