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| {{for|quotient spaces in linear algebra|quotient space (linear algebra)}}
| | Body Mass Index (BMI), sometimes known because the Quetelet Index of Obesity, is a formula utilized to determine the degree of weight of an individual. The formula was built in the nineteenth century by Adolphe Quetelet plus became internationally utilized to measure obesity in the 1980s. It continues to be widely used now despite being pretty inaccurate.<br><br>Then, as for calorie intake, a healthy individual must have 2,000 calories for an entire day of eating. Surprisingly, that's easy to do without starving yourself. Many persons over eat, when they will be full on a quarter of what exactly is consumed. If you eat correct portion models plus create healthy snack choices it could be easy!<br><br>Total body fat percentage consists of fat imperative to the metabolism plus stored fat. Essential fat is indispensable in terms of sustaining existence and reproductive functions of the body. The imperative percentage is better for women than which for guys considering their bodies have to deal with all the demands of childbearing plus other hormonal functions. According to the accepted norms, important fat is 2-5% for men and 10-13% for females. The minimum total body fat percentage recommended by healthcare experts is much more than this essential fat percentage. Storage fat refers to the fat accumulation inside adipose tissue. Some of the adipose tissue is imperative for safeguarding internal organs inside the chest and abdomen and it is actually somewhat higher for females than for men.<br><br>A [http://safedietplansforwomen.com/bmi-calculator bmi calculator females] is important tool to have specifically for females since they are more prone to get fat deposits than men. Hormonal changes are the most significant reason. Women are moreover proven to emotionally devour food than guy. When females are depressed, happy or tired they tend to look for comfort foods. Women are structurally different than guy to accommodate the growing baby inside the abdominal area. Plus a healthy female (21-36%) would have twice more fat compared to a healthy male (8-25%). Since women begin off with a better fat percentage than men, then it's not surprising that they are more liable to be overweight.<br><br>A person having a large frame and/or a great deal of muscle is flagged by Body Mass Index as being overweight--even whenever he has low body fat. This was true for disgraced cyclist Lance Armstrong at one point inside his athletic career.<br><br>Many sites might boast that is a individual has a high BMI, he or she should lose fat. This really is not true plus can occasionally result is severe health risks. Always consult an actual doctor when you're not sure. Never take the term of a business striving to market their product!<br><br>There are a lot of free premade diet programs plus tools which let you to create a diet online. I found the six sites indexed above to have the greatest free tools plus programs on the internet. All of these sites contain the tools and diet programs themselves so you do not have to click by links. I found Spark People to become the right free online diet from all sites I viewed. The site has everything we can require in a diet and every aspect is created to function with one another. So, go ahead, check out all these websites for yourself, and choose which is right for you. All which you have to bring is a little will energy plus you'll be able to mark 1 resolution, fat reduction, off a New Year's goals for advantageous! |
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| [[Image:Disk_to_Sphere_using_Quotient_Space.gif|thumb|Illustration of quotient space, [[N-sphere|{{math|''S''<sup>2</sup>}}]], obtained by ''gluing'' the boundary (in blue) of the [[Disk (mathematics)|disk]] {{math|''D''<sup>2</sup>}} together to a single point.]]
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| In [[topology]] and related areas of [[mathematics]], a '''quotient space''' (also called an '''identification space''') is, intuitively speaking, the result of identifying or "gluing together" certain points of a given [[topological space]]. The points to be identified are specified by an [[equivalence relation]]. This is commonly done in order to construct new spaces from given ones. The '''quotient topology''' consists of all sets with an [[open set|open]] [[preimage]] under the [[canonical projection map]] that maps each element to its equivalence class.
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| == Definition ==
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| Let {{math|(X,τ<sub>X</sub>)}} be a [[topological space]], and let ~ be an [[equivalence relation]] on ''X''. The '''quotient space''', <math>Y = X/\!\!\sim</math> is defined to be the set of [[equivalence class]]es of elements of {{math|''X''}}:
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| :<math>Y = \{ [x] : x \in X \} = \{\{v \in X : v \sim x\} : x \in X\},</math>
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| equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in ''X'':
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| :<math>\tau_Y = \left\{ U \subseteq Y : \bigcup U = \left( \bigcup_{ [a] \in U} [a] \right) \in \tau_X \right\}.</math>
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| Equivalently, we can define them to be those sets with an open preimage under the quotient map <math>q: X \to X/\!\!\sim</math> which sends a point in ''X'' to the equivalence class containing it:
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| :<math>\tau_Y=\left\{ U \subseteq Y : q^{-1}(U) \in \tau_X \right\}.</math>
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| The quotient topology is the [[final topology]] on the quotient space with respect to the quotient map. | |
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| == Examples ==
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| *'''Gluing.''' Often, topologists talk of gluing points together. If ''X'' is a topological space and points <math>x,y \in X</math> are to be "glued", then what is meant is that we are to consider the quotient space obtained from the equivalence relation ''a ~ b'' if and only if ''a = b'' or ''a = x, b = y'' (or ''a = y, b = x'').
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| *Consider the unit square ''I''<sup>2</sup> = [0,1]×[0,1] and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then ''I''<sup>2</sup>/~ is [[homeomorphic]] to the [[sphere|unit sphere]] ''S''<sup>2</sup>.
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| *'''[[Adjunction space]]'''. More generally, suppose ''X'' is a space and ''A'' is a [[subspace (topology)|subspace]] of ''X''. One can identify all points in ''A'' to a single equivalence class and leave points outside of ''A'' equivalent only to themselves. The resulting quotient space is denoted ''X''/''A''. The 2-sphere is then homeomorphic to the unit disc with its boundary identified to a single point: <math>D^{2}/\partial{D^{2}}</math>.
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| *Consider the set <math>X = \mathbb{R}</math> of all [[real number]]s with the ordinary topology, and write ''x'' ~ ''y'' [[if and only if]] ''x''−''y'' is an [[integer]]. Then the quotient space ''X''/~ is [[homeomorphic]] to the [[unit circle]] ''S''<sup>1</sup> via the homeomorphism which sends the equivalence class of ''x'' to exp(2π''ix'').
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| *A vast generalization of the previous example is the following: Suppose a [[topological group]] ''G'' [[group action|acts]] continuously on a space ''X''. One can form an equivalence relation on ''X'' by saying points are equivalent if and only if they lie in the same [[orbit (group theory)|orbit]]. The quotient space under this relation is called the '''orbit space''', denoted ''X''/''G''. In the previous example ''G'' = '''Z''' acts on '''R''' by translation. The orbit space '''R'''/'''Z''' is homeomorphic to ''S''<sup>1</sup>.
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| ''Warning'': The notation '''R'''/'''Z''' is somewhat ambiguous. If '''Z''' is understood to be a group acting on '''R''' then the [[Quotient group|quotient]] is the circle. However, if '''Z''' is thought of as a subspace of '''R''', then the quotient is an infinite [[bouquet of circles]] joined at a single point.
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| == Properties ==
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| Quotient maps {{nowrap|''q'' : ''X'' → ''Y''}} are characterized among surjective maps by the following property: if ''Z'' is any topological space and {{nowrap|''f'' : ''Y'' → ''Z''}} is any function, then ''f'' is continuous if and only if {{nowrap|''f'' ∘ ''q''}} is continuous.
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| [[Image:QuotientSpace-01.svg|center|Characteristic property of the quotient topology]]
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| The quotient space ''X''/~ together with the quotient map {{nowrap|''q'' : ''X'' → ''X''/~}} is characterized by the following [[universal property]]: if {{nowrap|''g'' : ''X'' → ''Z''}} is a continuous map such that {{nowrap|''a'' ~ ''b''}} implies {{nowrap|1=''g''(''a'') = ''g''(''b'')}} for all ''a'' and ''b'' in ''X'', then there exists a unique continuous map {{nowrap|''f'' : ''X''/~ → ''Z''}} such that {{nowrap|1=''g'' = ''f'' ∘ ''q''}}. We say that ''g'' ''descends to the quotient''.
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| The continuous maps defined on ''X''/~ are therefore precisely those maps which arise from continuous maps defined on ''X'' that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly used when studying quotient spaces. | |
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| Given a continuous surjection {{nowrap|''f'' : ''X'' → ''Y''}} it is useful to have criteria by which one can determine if ''f'' is a quotient map. Two sufficient criteria are that ''f'' be [[open map|open]] or [[closed map|closed]]. Note that these conditions are only [[sufficient condition|sufficient]], not [[necessary condition|necessary]]. It is easy to construct examples of quotient maps that are neither open nor closed.
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| == Compatibility with other topological notions ==
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| * [[Separation axioms|Separation]]
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| ** In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of ''X'' need not be inherited by ''X''/~, and ''X''/~ may have separation properties not shared by ''X''.
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| ** ''X''/~ is a [[T1 space]] if and only if every equivalence class of ~ is closed in ''X''.
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| ** If the quotient map is [[open map|open]], then ''X''/~ is a [[Hausdorff space]] if and only if ~ is a closed subset of the [[product space]] ''X''×''X''.
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| * [[Connectedness]]
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| ** If a space is connected or [[path connected]], then so are all its quotient spaces.
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| ** A quotient space of a [[simply connected]] or [[contractible]] space need not share those properties.
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| * [[Compact space|Compactness]]
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| ** If a space is compact, then so are all its quotient spaces.
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| ** A quotient space of a [[locally compact]] space need not be locally compact.
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| * [[Dimension]]
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| ** The [[topological dimension]] of a quotient space can be more (as well as less) than the dimension of the original space; [[space-filling curve]]s provide such examples.
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| ==See also==
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| ===Topology===
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| *[[Topological space]]
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| *[[Subspace (topology)]]
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| *[[Product space]]
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| *[[Disjoint union (topology)]]
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| *[[Final topology]]
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| *[[Mapping cone]]
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| ===Algebra===
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| *[[Quotient group]]
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| *[[Quotient space (linear algebra)]]
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| *[[Quotient category]]
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| *[[Mapping cone (homological algebra)]]
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| ==References==
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| *{{Cite book |first=Stephen |last=Willard |title=General Topology |year=1970 |publisher=Addison-Wesley |location=Reading, MA |isbn=0-486-43479-6 }}
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| * {{planetmath reference|id=2930|title=Quotient space}}
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| {{Reflist}}
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| [[Category:Topology]]
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| [[Category:General topology]]
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| [[Category:Group actions]]
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Body Mass Index (BMI), sometimes known because the Quetelet Index of Obesity, is a formula utilized to determine the degree of weight of an individual. The formula was built in the nineteenth century by Adolphe Quetelet plus became internationally utilized to measure obesity in the 1980s. It continues to be widely used now despite being pretty inaccurate.
Then, as for calorie intake, a healthy individual must have 2,000 calories for an entire day of eating. Surprisingly, that's easy to do without starving yourself. Many persons over eat, when they will be full on a quarter of what exactly is consumed. If you eat correct portion models plus create healthy snack choices it could be easy!
Total body fat percentage consists of fat imperative to the metabolism plus stored fat. Essential fat is indispensable in terms of sustaining existence and reproductive functions of the body. The imperative percentage is better for women than which for guys considering their bodies have to deal with all the demands of childbearing plus other hormonal functions. According to the accepted norms, important fat is 2-5% for men and 10-13% for females. The minimum total body fat percentage recommended by healthcare experts is much more than this essential fat percentage. Storage fat refers to the fat accumulation inside adipose tissue. Some of the adipose tissue is imperative for safeguarding internal organs inside the chest and abdomen and it is actually somewhat higher for females than for men.
A bmi calculator females is important tool to have specifically for females since they are more prone to get fat deposits than men. Hormonal changes are the most significant reason. Women are moreover proven to emotionally devour food than guy. When females are depressed, happy or tired they tend to look for comfort foods. Women are structurally different than guy to accommodate the growing baby inside the abdominal area. Plus a healthy female (21-36%) would have twice more fat compared to a healthy male (8-25%). Since women begin off with a better fat percentage than men, then it's not surprising that they are more liable to be overweight.
A person having a large frame and/or a great deal of muscle is flagged by Body Mass Index as being overweight--even whenever he has low body fat. This was true for disgraced cyclist Lance Armstrong at one point inside his athletic career.
Many sites might boast that is a individual has a high BMI, he or she should lose fat. This really is not true plus can occasionally result is severe health risks. Always consult an actual doctor when you're not sure. Never take the term of a business striving to market their product!
There are a lot of free premade diet programs plus tools which let you to create a diet online. I found the six sites indexed above to have the greatest free tools plus programs on the internet. All of these sites contain the tools and diet programs themselves so you do not have to click by links. I found Spark People to become the right free online diet from all sites I viewed. The site has everything we can require in a diet and every aspect is created to function with one another. So, go ahead, check out all these websites for yourself, and choose which is right for you. All which you have to bring is a little will energy plus you'll be able to mark 1 resolution, fat reduction, off a New Year's goals for advantageous!