Humidity

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 Hausdorff space βX. The Stone–Čech compactification βX 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 βX (in a unique way). If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX. For general topological spaces X, the map from X to βX need not be injective.

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 βX often remains elusive. In particular, proofs that βN \ N is nonempty do not give an explicit description of any particular point in βN \ N.

The Stone–Čech compactification occurs implicitly in a paper by Template:Harvs and was given explicitly by Template:Harvs and Template:Harvs.

Universal property and functoriality

βX 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 βf: βX → K.

As is usual for universal properties, this universal property, together with the fact that βX is a compact Hausdorff space containing X, characterizes βX up to homeomorphism.

Some authors add the assumption that the starting space be Tychonoff (or even locally compact Hausdorff), for the following reasons:

• The map from X to its image in βX is a homeomorphism if and only if X is Tychonoff.
• The map from X to its image in βX is a homeomorphism to an open subspace if and only if X is locally compact Hausdorff.

The Stone–Čech construction can be performed for more general spaces X, but the map X → βX need not be a homeomorphism to the image of X (and sometimes is not even injective).

The extension property makes β a functor from Top (the 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 βX 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 βX). i.e. Hom(βX, K) = Hom(X, UK), which means that β is left adjoint to U. This implies that CHaus is a reflective subcategory of Top with reflector β.

Constructions

Construction using products

One attempt to construct the Stone–Čech compactification of X is to take the closure of the image of X in

$\prod C$

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.

Construction using the unit interval

One way of constructing βX is to consider the map

$X\to [0,1]^{C}$

$x\mapsto (f(x))_{f\in C}$

where C is the set of all continuous functions from X into [0, 1]. This may be seen to be a continuous map onto its image, if [0, 1]^C is given the product topology. By Tychonoff's theorem we have that [0, 1]^C is compact since [0, 1] is. Consequently, the closure of X in [0, 1]^C is a compactification of X.

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]^C. 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.

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.

Construction using ultrafilters

Alternatively, if X is discrete, one can construct βX as the set of all ultrafilters on X, with a topology known as Stone topology. The elements of X correspond to the principal ultrafilters.

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 βf(F) = x. This may be verified to be a continuous extension of f.

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.

The construction can be generalized to arbitrary Tychonoff spaces by using maximal filters of zero sets instead of ultrafilters. (Filters of closed sets suffice if the space is normal.)

Construction using C*-algebras

In case X is a completely regular Hausdorff space, the Stone–Čech compactification can be identified with the spectrum of C_b(X). Here C_b(X) denotes the C*-algebra of all continuous bounded functions on X with sup-norm. Notice that C_b(X) is the multiplier algebra of C₀(X).

The Stone–Čech compactification of the natural numbers

In the case where X is locally compact, e.g. N or R, the image of X forms an open subset of βX, 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, βX \ X. This is a closed subset of βX, and so is compact. We consider N with its discrete topology and write βN \ N = N* (but this does not appear to be standard notation for general X).

One can view βN as the set of ultrafilters on N, with the topology generated by sets of the form $\{F:U\in F\}$ for U a subset of N. The set N corresponds to the set of principal ultrafilters, and the set N* to the set of free ultrafilters.

The easiest way to see this is isomorphic to βN 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 βf(F) = x. This may readily be verified to be a continuous extension.

(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.)

The study of βN, 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.

These state:

• Every compact Hausdorff space of weight at most $\aleph _{1}$ (see Aleph number) is the continuous image of N* (this does not need the continuum hypothesis, but is less interesting in its absence).
• If the continuum hypothesis holds then N* is the unique Parovicenko space, up to isomorphism.

These were originally proved by considering Boolean algebras and applying Stone duality.

Jan van Mill has described βN 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). It has relatively recently been observed that this characterisation isn't quite right - there is in fact a fourth head of βN, in which forcing axioms and Ramsey type axioms give properties of βN 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*)² ≠ N*, while the continuum hypothesis implies the opposite.

An application: the dual space of the space of bounded sequences of reals

The Stone–Čech compactification βN can be used to characterize ℓ^∞(N) (the Banach space of all bounded sequences in the scalar field R or C, with supremum norm) and its dual space.

Given a bounded sequence a in ℓ^∞(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 βa: βN → B. This extension does not depend on the ball B we consider.

We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over βN.

$l^{\infty }(\mathbf {N} )\to C(\beta \mathbf {N} )$

This map is bijective since every function in C(βN) must be bounded and can then be restricted to a bounded scalar sequence.

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).

Thus, ℓ^∞(N) can be identified with C(βN). This allows us to use the Riesz representation theorem and find that the dual space of ℓ^∞(N) can be identified with the space of finite Borel measures on βN.

Finally, it should be noticed that this technique generalizes to the L^∞ space of an arbitrary measure space X. However, instead of simply considering the space βX 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^∞(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).

Addition on the Stone–Čech compactification of the naturals

The natural numbers form a monoid under addition. It turns out that this operation can be extended (in more than one way) to βN, turning this space also into a monoid, though rather surprisingly a non-commutative one.

For any subset, A, of N and a positive integer n in N, we define

$A-n=\{k\in \mathbf {N} \mid k+n\in A\}.$

Given two ultrafilters F and G on N, we define their sum by

$F+G={\Big \{}A\subset \mathbf {N} \mid \{n\in \mathbf {N} \mid A-n\in F\}\in G{\Big \}};$

it can be checked that this is again an ultrafilter, and that the operation + is associative (but not commutative) on βN and extends the addition on N; 0 serves as a neutral element for the operation + on βN. The operation is also right-continuous, in the sense that for every ultrafilter F, the map

$\beta \mathbf {N} \to \beta \mathbf {N}$

$G\mapsto F+G$

is continuous.

See also

• One-point compactification
• Wallman compactification
• Corona set of a space, the complement of its image in the Stone–Čech compactification.

Notes

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References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Template:PlanetMath
• Dror Bar-Natan, Ultrafilters, Compactness, and the Stone–Čech compactification