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| In the mathematical discipline of [[general topology]], '''Stone–Čech compactification''' is a technique for constructing a universal map from a topological space ''X'' to a [[Compact space|compact]] [[Hausdorff space]] β''X''. The Stone–Čech compactification β''X'' of a topological space ''X'' is the largest compact Hausdorff space "generated" by ''X'', in the sense that any map from ''X'' to a compact Hausdorff space factors through β''X'' (in a unique way). If ''X'' is a [[Tychonoff space]] then the map from ''X'' to its image in β''X'' is a homeomorphism, so ''X'' can be thought of as a (dense) subspace of β''X''. For general topological spaces ''X'', the map from ''X'' to β''X'' need not be injective.
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| A form of the [[axiom of choice]] is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces ''X'', an accessible concrete description of β''X'' often remains elusive. In particular, proofs that β'''N''' \ '''N''' is nonempty do not give an explicit description of any particular point in β'''N''' \ '''N'''.
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| The Stone–Čech compactification occurs implicitly in a paper by {{harvs|txt|authorlink=Andrey Nikolayevich Tikhonov|last=Tychonoff|year=1930}} and was given explicitly by {{harvs|authorlink=Marshall Stone|first=Marshall|last=Stone|year=1937|txt=yes}} and {{harvs|authorlink=Eduard Čech|first=Eduard |last=Čech|year=1937|txt=yes}}.
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| == Universal property and functoriality ==
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| β''X'' is a compact Hausdorff space together with a continuous map from ''X'' and has the following [[universal property]]: any [[continuous map]] ''f:'' ''X'' → ''K'', where ''K'' is a compact Hausdorff space, lifts uniquely to a continuous map β''f:'' β''X'' → ''K''.
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| :[[Image:Stone–Cech compactification.png|120px]]
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| As is usual for universal properties, this universal property, together with the fact that β''X'' is a compact Hausdorff space containing ''X'', characterizes β''X'' [[up to]] [[homeomorphism]].
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| Some authors add the assumption that the starting space be Tychonoff (or even locally compact Hausdorff), for the following reasons:
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| *The map from ''X'' to its image in β''X'' is a homeomorphism if and only if ''X'' is Tychonoff.
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| *The map from ''X'' to its image in β''X'' is a homeomorphism to an open subspace if and only if ''X'' is locally compact Hausdorff.
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| The Stone–Čech construction can be performed for more general spaces ''X'', but the map ''X'' → β''X'' need not be a homeomorphism to the image of ''X'' (and sometimes is not even injective).
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| The extension property makes β a [[functor]] from '''Top''' (the [[Category (mathematics)|category]] of topological spaces) to '''CHaus''' (the category of compact Hausdorff spaces). If we let ''U'' be the [[inclusion functor]] from '''CHaus''' into '''Top''', maps from β''X'' to ''K'' (for ''K'' in '''CHaus''') correspond bijectively to maps from ''X'' to ''UK'' (by considering their restriction to ''X'' and using the universal property of β''X''). i.e. Hom(β''X'', ''K'') = Hom(''X'', ''UK''), which means that β is [[adjoint functor|left adjoint]] to ''U''. This implies that '''CHaus''' is a [[reflective subcategory]] of '''Top''' with reflector β.
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| ==Constructions==
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| ===Construction using products===
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| One attempt to construct the Stone–Čech compactification of ''X'' is to take the closure of the image of ''X'' in
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| :<math>\prod C</math>
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| where the product is over all maps from ''X'' to compact Hausdorff spaces ''C''. This works intuitively but fails for the technical reason that the collection of all such maps is a proper class rather than a set. There are several ways to modify this idea to make it work; for example, one can restrict the compact Hausdorff spaces ''C'' to have underlying set ''P''(''P''(''X'')) (the power set of the power set of ''X''), which is sufficiently large that it has cardinality at least equal to that of every compact Hausdorff set to which ''X'' can be mapped with dense image.
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| ===Construction using the unit interval===
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| One way of constructing β''X'' is to consider the map
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| :<math>X \to [0, 1]^{C}</math>
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| :<math>x \mapsto ( f(x) )_{f \in C}</math>
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| where ''C'' is the set of all [[continuous function]]s from ''X'' into [0, 1]. This may be seen to be a continuous map onto its image, if [0, 1]<sup>''C''</sup> is given the [[product topology]]. By [[Tychonoff's theorem]] we have that [0, 1]<sup>''C''</sup> is compact since [0, 1] is. Consequently, the closure of ''X'' in [0, 1]<sup>''C''</sup> is a compactification of ''X''.
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| In fact, this closure is the Stone–Čech compactification. To verify this, we just need to verify that the closure satisfies the appropriate universal property. We do this first for ''K'' = [0, 1], where the desired extension of ''f:'' ''X'' → [0, 1] is just the projection onto the ''f'' coordinate in [0, 1]<sup>''C''</sup>. In order to then get this for general compact Hausdorff ''K'' we use the above to note that ''K'' can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions.
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| The special property of the unit interval needed for this construction to work is that it is a '''cogenerator''' of the category of compact Hausdorff spaces: this means that if ''A'' and ''B'' are compact Hausdorff spaces, and ''f'' and ''g'' are distinct maps from ''A'' to ''B'', then there is a map ''h'' from ''B'' to [0, 1] such that ''hf'' and ''hg'' are distinct. Any other cogenerator (or cogenerating set) can be used in this construction.
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| ===Construction using ultrafilters===
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| Alternatively, if ''X'' is discrete, one can construct β''X'' as the set of all [[ultrafilter]]s on ''X'', with a topology known as ''Stone topology''. The elements of ''X'' correspond to the [[ultrafilter|principal ultrafilter]]s.
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| Again we verify the universal property: For ''f:'' ''X'' → ''K'' with ''K'' compact Hausdorff and ''F'' an ultrafilter on ''X'' we have an ultrafilter ''f(F)'' on ''K''. This has a unique limit because ''K'' is compact, say ''x'', and we define β''f(F)'' = ''x''. This may be verified to be a continuous extension of ''f''.
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| Equivalently, one can take the [[Stone space]] of the [[complete Boolean algebra]] of all subsets of ''X'' as the Stone–Čech compactification. This is really the same construction, as the Stone space of this Boolean algebra is the set of ultrafilters (or equivalently prime ideals, or homomorphisms to the 2 element Boolean algebra) of the Boolean algebra, which is the same as the set of ultrafilters on ''X''.
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| The construction can be generalized to arbitrary Tychonoff spaces by using maximal filters of [[zero set]]s instead of ultrafilters. (Filters of closed sets suffice if the space is normal.)
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| ===Construction using C*-algebras===
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| In case ''X'' is a completely regular Hausdorff space, the Stone–Čech compactification can be identified with the [[Spectrum of a C*-algebra|spectrum]] of C<sub>b</sub>(''X''). Here C<sub>b</sub>(''X'') denotes the [[C*-algebra]] of all continuous bounded functions on ''X'' with sup-norm. Notice that C<sub>b</sub>(''X'') is the [[multiplier algebra]] of C<sub>0</sub>(''X'').
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| ==The Stone–Čech compactification of the natural numbers==
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| In the case where ''X'' is [[locally compact]], e.g. '''N''' or '''R''', the image of ''X'' forms an open subset of β''X'', or indeed of any compactification, (this is also a necessary condition, as an open subset of a compact Hausdorff space is locally compact). In this case one often studies the remainder of the space, β''X'' \ ''X''. This is a closed subset of β''X'', and so is compact. We consider '''N''' with its [[discrete topology]] and write β'''N''' \ '''N''' = '''N'''* (but this does not appear to be standard notation for general ''X'').
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| One can view β'''N''' as the set of [[ultrafilter]]s on '''N''', with the topology generated by sets of the form <math>\{ F : U \in F \}</math> for ''U'' a subset of '''N'''. The set '''N''' corresponds to the set of [[ultrafilter|principal ultrafilter]]s, and the set '''N'''* to the set of [[ultrafilter|free ultrafilters]].
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| The easiest way to see this is isomorphic to β'''N''' is to show that it satisfies the universal property. For ''f:'' '''N''' → ''K'' with ''K'' compact Hausdorff and ''F'' an ultrafilter on '''N''' we have an ultrafilter ''f(F)'' on ''K'', the pushforward of ''F''. This has a unique limit, say ''x'', because ''K'' is compact Hausdorff, and we define β''f''(''F'') = ''x''. This may readily be verified to be a continuous extension.
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| (A similar but slightly more involved construction of the Stone–Čech compactification as a set of certain maximal filters can also be given for a general Tychonoff space ''X''.)
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| The study of β'''N''', and in particular '''N'''*, is a major area of modern [[set-theoretic topology]]. The major results motivating this are [[Parovicenko's theorems]], essentially characterising its behaviour under the assumption of the [[continuum hypothesis]].
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| These state:
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| * Every compact Hausdorff space of [[weight of a space|weight]] at most <math>\aleph_1</math> (see [[Aleph number]]) is the continuous image of '''N'''* (this does not need the continuum hypothesis, but is less interesting in its absence).
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| * If the continuum hypothesis holds then '''N'''* is the unique [[Parovicenko space]], up to isomorphism.
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| These were originally proved by considering [[Boolean algebra (structure)|Boolean algebra]]s and applying [[Stone duality]].
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| Jan van Mill has described β'''N''' as a 'three headed monster' — the three heads being a smiling and friendly head (the behaviour under the assumption of the continuum hypothesis), the ugly head of independence which constantly tries to confuse you (determining what behaviour is possible in different models of set theory), and the third head is the smallest of all (what you can prove about it in [[ZFC]]).<ref>{{Citation
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| | first = Jan
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| | last = van Mill
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| | editor-last = Kunen
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| | editor-first = Kenneth
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| | editor2-last = Vaughan
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| | editor2-first = Jerry E.
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| | contribution = An introduction to βω
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| | title = Handbook of Set-Theoretic Topology
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| | year = 1984
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| | pages = 503–560
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| | publisher = North-Holland
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| | isbn = 0-444-86580-2}}</ref> It has relatively recently been observed that this characterisation isn't quite right - there is in fact a fourth head of β'''N''', in which [[forcing (mathematics)|forcing axioms]] and Ramsey type axioms give properties of β'''N''' almost diametrically opposed to those under the continuum hypothesis, giving very few maps from '''N'''* indeed. Examples of these axioms include the combination of [[Martin's axiom]] and the [[Open colouring axiom]] which, for example, prove that ('''N'''*)<sup>2</sup> ≠ '''N'''*, while the continuum hypothesis implies the opposite.
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| === An application: the dual space of the space of bounded sequences of reals ===
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| The Stone–Čech compactification β'''N''' can be used to characterize ℓ<sup>∞</sup>('''N''') (the [[Banach space]] of all bounded sequences in the scalar field '''R''' or '''C''', with [[supremum norm]]) and its [[dual space]].
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| Given a bounded sequence ''a'' in ℓ<sup>∞</sup>('''N'''), there exists a closed ball ''B'' that contains the image of ''a'' (''B'' is a subset of the scalar field). ''a'' is then a function from '''N''' to ''B''. Since '''N''' is discrete and ''B'' is compact and Hausdorff, ''a'' is continuous. According to the universal property, there exists a unique extension β''a:'' β'''N''' → ''B''. This extension does not depend on the ball ''B'' we consider.
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| We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over β'''N'''.
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| :<math> l^\infty(\mathbf{N}) \to C(\beta \mathbf{N}) </math>
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| This map is bijective since every function in C(β'''N''') must be bounded and can then be restricted to a bounded scalar sequence.
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| If we further consider both spaces with the sup norm the extension map becomes an isometry. Indeed, if in the construction above we take the smallest possible ball ''B'', we see that the sup norm of the extended sequence does not grow (although the image of the extended function can be bigger).
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| Thus, ℓ<sup>∞</sup>('''N''') can be identified with C(β'''N'''). This allows us to use the [[Riesz representation theorem]] and find that the dual space of ℓ<sup>∞</sup>('''N''') can be identified with the space of finite [[Borel measure]]s on β'''N'''.
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| Finally, it should be noticed that this technique generalizes to the ''L''<sup>∞</sup> space of an arbitrary [[measure space]] ''X''. However, instead of simply considering the space β''X'' of ultrafilters on ''X'', the right way to generalize this construction is to consider the [[Stone space]] ''Y'' of the measure algebra of ''X'': the spaces ''C''(''Y'') and ''L''<sup>∞</sup>(''X'') are isomorphic as C*-algebras as long as ''X'' satisfies a reasonable finiteness condition (that any set of positive measure contains a subset of finite positive measure).
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| ===Addition on the Stone–Čech compactification of the naturals===
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| The natural numbers form a [[monoid]] under [[addition]]. It turns out that this operation can be extended (in more than one way) to β'''N''', turning this space also into a monoid, though rather surprisingly a non-commutative one.
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| For any subset, ''A'', of '''N''' and a positive integer ''n'' in '''N''', we define
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| :<math>A-n=\{k\in\mathbf{N}\mid k+n\in A\}.</math>
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| Given two ultrafilters ''F'' and ''G'' on '''N''', we define their sum by
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| :<math>F+G = \Big\{A\subset\mathbf{N}\mid \{n\in\mathbf{N}\mid A-n\in F\}\in G\Big\};</math>
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| it can be checked that this is again an ultrafilter, and that the operation + is [[associative]] (but not commutative) on β'''N''' and extends the addition on '''N'''; 0 serves as a neutral element for the operation + on β'''N'''. The operation is also right-continuous, in the sense that for every ultrafilter ''F'', the map
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| :<math>\beta \mathbf{N}\to\beta \mathbf{N}</math>
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| :<math>G \mapsto F+G</math>
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| is continuous.
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| ==See also==
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| *[[One-point compactification]]
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| *[[Wallman compactification]]
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| *[[Corona set]] of a space, the complement of its image in the Stone–Čech compactification.
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| == Notes ==
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| {{Reflist}}
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| == References ==
| |
| *{{citation|first=E.|last= Čech|title=On bicompact spaces|journal= Ann. Math. |volume= 38 |year=1937 |pages= 823–844
| |
| |doi=10.2307/1968839|issue=4|publisher=The Annals of Mathematics, Vol. 38, No. 4 |jstor=1968839}}
| |
| *{{citation
| |
| |last=Hindman|first= Neil|last2= Strauss|first2= Dona
| |
| |title=Algebra in the Stone-Cech compactification. Theory and applications |series=de Gruyter Expositions in Mathematics|volume= 27|publisher= Walter de Gruyter & Co.|publication-place= Berlin|year= 1998|pages= xiv+485 pp. |isbn= 3-11-015420-X
| |
| |mr=1642231}}
| |
| *{{springer|id=S/s090340|first=I.G. |last=Koshevnikova}}
| |
| *{{citation|first=M.H.|last= Stone|title=Applications of the theory of Boolean rings to general topology |journal=Trans. Amer. Soc. |volume= 41 |year=1937|pages= 375–481
| |
| |issue=3|doi=10.2307/1989788|publisher=Transactions of the American Mathematical Society, Vol. 41, No. 3 |jstor=1989788}}
| |
| *{{Citation | last1=Tychonoff | first1=A. | title=Über die topologische Erweiterung von Räumen | url=http://dx.doi.org/10.1007/BF01782364 | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01782364 | year=1930 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=102 | pages=544–561}}
| |
| *{{citation|first=Allen|last=Shields|title=Years ago|journal=The Mathematical Intelligencer|volume= 9|issue=2 |year=1987 |pages= 61–63|doi=10.1007/BF03025901}}
| |
| | |
| ==External links==
| |
| * {{PlanetMath|title=Stone-Čech compactification}}
| |
| * Dror Bar-Natan, ''[http://www.math.toronto.edu/~drorbn/classes/9293/131/ultra.pdf Ultrafilters, Compactness, and the Stone–Čech compactification]''
| |
| | |
| {{DEFAULTSORT:Stone-Cech compactification}}
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| [[Category:General topology]]
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| [[Category:Compactification]]
| |
If you are reading this article, you need to anticipate to make some modifications to your diet plan. That's excellent media indeed! With all that facts available, figuring out how to start can often be difficult. Don't stress, and look for these nutrients ideas! Right here, there are actually tips that may assist you in following a healthy diet.
Prepared food items come with a nutritious content label. Get to know these labeling and learn how to study them properly. Numerous bad elements, including salt and sugars, might be camouflaging behind that "lessened-excess fat" tag. Processed foods are almost never a wholesome choice. The content label need to collection ingredients that are standard enough for anyone to know. Try to steer clear of food items that have lots of artificial substances.
Start out with tiny techniques when attemping to improve your diet program. This isn't one thing that needs to be hurried. This can jolt and confound the body, which you do not want. Take into consideration seeking more healthy options the next month approximately, and you may then made a serious excellent practice of much better consuming.
Including garlic for your foods has numerous benefits. Garlic cloves has been recognized as an effective folk treatment for combating disease. The latest research has pointed out it is useful in avoiding heart problems and cancers. Garlic cloves even offers contra --yeast and anti-bacterial consequences to assist your body organs. With every one of these wonderful abilities, garlic, whether for an get or even in clove type, is a vital every single day food.
The stealthy method is by adding more healthy products when cooking your normal foods. You could add nutrients in your children's meals employing a stealth way of including vegetables into dishes. Consider adding a certain amount of flax together with your fresh fruit shakes, or swap the soil meat with turkey within your chili. Your picky eaters will consume more healthy food items while observing a minimum of alter.
Take note of simply how much you will be ingesting daily. Overeating foods can overnourish your body, which doesn't feel totally excellent and might make you retail store excess fat. Way too many unhealthy calories will damage your state of health.
Eat meals for each shade and often. Fruits and vegetables that happen to be dazzling in color are typically low-calorie and nutritional abundant. You ought to be planning vegetables with every meal and eating some fruits like a snack food. Skin contain anti-oxidant benefits, so take in these too when they are delicious.
Sugar should be operated to be able to live a healthy existence. A lot of people incorrectly feel that all juice refreshments are far healthier than soft drink. Strangely, some brands of juice are in fact a lot more sugary than soft drinks. As a result, it is vital to concentrate on the foodstuffs consume, specifically in relation to sugars content material.
Usually do not fall for labeling that say excess fat cost-free or no trans fat. Though they may possibly lack excess fat, they frequently include substantial amounts of glucose. It is recommended to have a close look at the information on the content label before choosing the piece.
Eating healthy might not be effortless, but the more clean your daily diet is, the higher you can expect to appear and feel. A terrific way to look after on your own is as simple as reducing your enhanced sugars intake. Be suspicious of things like fresh fruit juices and sodas. These things consist of a higher volume of glucose. You would like to avoid sugars. When you decrease the all kinds of sugar in what you eat, you can expect to loose excess weight speedier. You'll also sense and appear far better.
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To keep better diet, make certain meals are nutritionally well balanced appropriately. Our bodies requires vitamins and minerals, vitamin supplements, amino acids, and fatty acids to function nicely. When this sort of nutrients and vitamins can be bought in nutritional supplements, our body is much better capable of extract them from meals sources.
Omitting morning meal is unfavorable to any successful diet regime. It is a frequent perception that omitting your morning meal can reduced the daily consumption of calories. But through the elimination of this significant dinner, you have a danger for being so starving afterwards that you simply overeat at the conclusion of the day. You may find yourself ingesting far more calorie consumption than if you had basically eaten a healthy your morning meal at the start of the day.
Scent foods like peppermints. Once you take in these types of food, the aroma will allow you to really feel far more happy. A lot of people think that these scents actually trick our bodies into convinced that the individual is eating or has eaten foods. In the event you aren't eager, you won't overindulge.
A great way to rejuvenate your daily diet-shifting development is usually to remember the factors that very first inspired you to definitely commence having much healthier. This may fluctuate for various men and women, however when you return to the basic principles, it can certainly help you to definitely recognize and re-calibrate oneself with regards to your targets are worried.
When a lot of people focus on diet, they regularly only consider foods consumption and then forget that beverages play a significant role too. Concentrate on the number of calories you will be placing into your body just with all the cocktails that you just consume. Lessen or get rid of cocktails like soft drink, drink and liquid drinks with extra sugars or higher-fructose corn syrup.
Eat small dishes with greater frequency. Having these smaller sized meals a couple of hours apart in between 5 and half a dozen periods everyday can help digestion and maintain the weight downward. Clearly a far healthier excess weight permits you to reduce the potential risk of coronary heart concerns and diabetic issues which can be one other reason to concentrate on your nutrients. This could minimize the cravings for unhealthy foods that you may have also.
So that you can have a healthy diet, there are some food items you must steer clear of having. Initial, sugars tend not to get a lean body and may be unfavorable. Second, food products that have gone through a bleaching approach, such as white colored bread or rice, do not possess as many nutrients and vitamins his or her dark brown alternatives. Make sure you get healthful quickly as an alternative to bad fats or trans fats.
Vegetables could be grown without chemicals without paying big money for these people. You are able to expand strawberries and tomato plants in storage containers, and outdoor patio growing plants can produce a shocking quantity of legumes and peppers, even lettuce.