Product of group subsets: Difference between revisions
en>Sandeepr.murthy No edit summary |
en>J.Gowers Corrected an inaccuracy |
||
| Line 1: | Line 1: | ||
In [[mathematics]], '''pointless topology''' (also called '''point-free''' or '''pointfree topology''') is an approach to [[topology]] that avoids mentioning points. The name 'pointless topology' is due to [[John von Neumann]].<ref>Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5 </ref> The ideas of pointless topology are closely related to [[mereotopology| mereotopologies]] in which regions (sets) are treated as foundational without explicit reference to underlying point sets. | |||
==General concepts== | |||
Traditionally, a [[topological space]] consists of a [[Set (mathematics)|set]] of [[point (topology)|points]], together with a system of [[open set]]s. These open sets with the operations of [[intersection (set theory)|intersection]] and [[union (set theory)|union]] form a [[lattice (order)|lattice]] with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the [[category theory|category]] of pointless topological spaces, also called [[Frames and locales|locales]], as an extension of the category of ordinary topological spaces. | |||
==Categories of frames and locales== | |||
:{{main|Frames and locales}} | |||
Formally, a '''frame''' is defined to be a [[lattice (order)|lattice]] ''L'' in which finite [[Meet (mathematics)|meet]]s [[Distributivity (order theory)|distribute]] over arbitrary [[Join (mathematics)|joins]], i.e. every (even infinite) subset {''a''<sub>i</sub>} of ''L'' has a [[supremum]] ⋁''a''<sub>''i''</sub> such that | |||
:<math>b \wedge \left( \bigvee a_i\right) = \bigvee \left(a_i \wedge b\right)</math> | |||
for all ''b'' in ''L''. These frames, together with lattice homomorphisms that respect arbitrary suprema, form a category. The [[dual (category theory)|dual]] of the '''category of frames''' is called the '''category of locales''' and generalizes the category '''[[category of topological spaces|Top]]''' of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every [[continuous function (topology)|continuous map]] between topological spaces ''X'' and ''Y'' induces a map between the lattices of open sets ''in the opposite direction'' as for every continuous function ''f'': ''X'' → ''Y'' and every open set ''O'' in ''Y'' the [[inverse image]] ''f''<sup> -1</sup>(''O'') is an open set in ''X''. | |||
==Relation to point-set topology== | |||
It is possible to translate most concepts of [[point-set topology]] into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the [[axiom of choice]], this is not true for some of their analogues in locale theory. This can be useful if one works in a [[topos]] that does not have the axiom of choice. | |||
The concept of "product of locales" diverges slightly from the concept of "[[Product_topology|product of topological spaces]]", and this divergence has been called a disadvantage of the locale approach. | |||
Others{{who|date=November 2010}} claim that the locale product is more natural, and point to several "desirable" properties{{Which?|date=November 2010}} not shared by products of topological spaces. | |||
For almost all spaces (more precisely for [[sober space]]s), the topological product and the localic product have the same set of points. The products differ in how equality between sets of open rectangles, the canonical base for the product topology, is defined: equality for the topological product means the same set of points is covered; | |||
equality for the localic product means provable equality using the frame axioms. As a result, two open sublocales of a localic product may contain exactly the same points without being equal. | |||
A point where locale theory and topology diverge much more strongly is the concept of subspaces vs. sublocales. | |||
The rational numbers have ''c'' subspaces but 2<sup>''c''</sup> sublocales. The proof for the latter statement is due to [[John Isbell]], and uses the fact that the rational numbers have ''c'' many pairwise almost disjoint (= finite intersection) closed subspaces.<ref>{{citation | |||
| last = Isbell | first = John | |||
| contribution = Some problems in descriptive locale theory | |||
| location = Providence, RI | |||
| mr = 1192150 | |||
| pages = 243–265 | |||
| publisher = Amer. Math. Soc. | |||
| series = CMS Conf. Proc. | |||
| title = Category theory 1991 (Montreal, PQ, 1991) | |||
| volume = 13 | |||
| year = 1992}}. See in particular [http://books.google.com/books?id=3hwiWc4iau0C&pg=PA245 p. 245].</ref> | |||
==See also== | |||
* [[Heyting algebra]]. A locale is a [[complete Heyting algebra]]. | |||
* Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between [[sober space]]s and spatial locales, are to be found in the article on [[Stone duality]]. | |||
* [[Point-free geometry]] | |||
* [[Mereology]] | |||
* [[Mereotopology]] | |||
==References== | |||
{{reflist}} | |||
*[[Peter Johnstone (mathematician)|Johnstone, Peter T.]], 1983, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014 The point of pointless topology,]" ''Bulletin of the American Mathematical Society 8(1)'': 41-53. | |||
[[Category:Category theory]] | |||
[[Category:General topology]] | |||
Revision as of 23:31, 27 November 2013
In mathematics, pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann.[1] The ideas of pointless topology are closely related to mereotopologies in which regions (sets) are treated as foundational without explicit reference to underlying point sets.
General concepts
Traditionally, a topological space consists of a set of points, together with a system of open sets. These open sets with the operations of intersection and union form a lattice with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.
Categories of frames and locales
- Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i.e. every (even infinite) subset {ai} of L has a supremum ⋁ai such that
for all b in L. These frames, together with lattice homomorphisms that respect arbitrary suprema, form a category. The dual of the category of frames is called the category of locales and generalizes the category Top of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every continuous map between topological spaces X and Y induces a map between the lattices of open sets in the opposite direction as for every continuous function f: X → Y and every open set O in Y the inverse image f -1(O) is an open set in X.
Relation to point-set topology
It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the axiom of choice, this is not true for some of their analogues in locale theory. This can be useful if one works in a topos that does not have the axiom of choice.
The concept of "product of locales" diverges slightly from the concept of "product of topological spaces", and this divergence has been called a disadvantage of the locale approach. OthersTemplate:Who claim that the locale product is more natural, and point to several "desirable" propertiesTemplate:Which? not shared by products of topological spaces.
For almost all spaces (more precisely for sober spaces), the topological product and the localic product have the same set of points. The products differ in how equality between sets of open rectangles, the canonical base for the product topology, is defined: equality for the topological product means the same set of points is covered; equality for the localic product means provable equality using the frame axioms. As a result, two open sublocales of a localic product may contain exactly the same points without being equal.
A point where locale theory and topology diverge much more strongly is the concept of subspaces vs. sublocales. The rational numbers have c subspaces but 2c sublocales. The proof for the latter statement is due to John Isbell, and uses the fact that the rational numbers have c many pairwise almost disjoint (= finite intersection) closed subspaces.[2]
See also
- Heyting algebra. A locale is a complete Heyting algebra.
- Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober spaces and spatial locales, are to be found in the article on Stone duality.
- Point-free geometry
- Mereology
- Mereotopology
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- Johnstone, Peter T., 1983, "The point of pointless topology," Bulletin of the American Mathematical Society 8(1): 41-53.
- ↑ Garrett Birkhoff, VON NEUMANN AND LATTICE THEORY, John Von Neumann 1903-1957, J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5
- ↑ 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. See in particular p. 245.