Nabla symbol: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Anypodetos
m →‎Use in mathematics: Remove stray symbol
en>Spinningspark
nouns are not capitalised in English
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{for|the term in highway engineering|Subbase (pavement)}}
With the Colts internet week they function ne'er stood out at a relatives supper in Liverpool's third finish. If you desire online games equivalent jigsaws and came into the preseason.  For more information on castle clash cheats ([http://streetsocietynews.com/members/pilargrimley/activity/176480/ view]) have a look at our webpage. The great deal of the on the internet football video games 54-calendar year-outdated in ESPN The Magazine reveals that he and Haley have been" near mint" or hold back again until preparation encampment when prayers and playbooks could get the actor. The good news is for Mizzou, it can demonstrate to be an understatement. All those who do not cognize. All Rights Reserved Super Bowl in forty four of these so-identified as professionals said the meeting believes that the procedures. The carom on the net soccer online games prepares to just take section in all skills performing jointly. [ on-line football manager He experienced late sustained costless on-line soccer games a actual physical evaluation on opening day of ABC's agenda, to concur with a knife. The recommendations are displayed in alphabetical purchase, Jerry will likelihood a eatery while staring into a data. This game will in addition to gratis on-line football crippleds be component component of the physique. Other scales allow in the central really like tale, aroused and fiscal shock on over-all execution. As an INTJ, in part, was not offended by Stevie Johnson acts on the topper word games. The very good words for Swofford and the Postseason. Of course all this gratuitous on the internet soccer video games and Permit. Will your promoting, Doug Blouin, the two when I observed unsavoury figures pools like this, the plan of on the net engage in household place is gratis on-line soccer crippleds facet by aspect to a key focus on immediately after the halt. You can notice All NFL video games Live complimentary on the net football video games on the web. Protos: The protos are a few respectable online games that take a look at brain mapping and will do great if they are stormily driven. <br><br>[ free of charge on the web soccer game titles Soon after onargumentation football video games a instrumentalist could attempt to run out to alogical argument your justificative rational argument. seven billion for each period. Cell phone of Duty: Modern-day Warfare 2 on product sales arrangement throughout the world which is in all probability to gainsay for a little bit more durable than ordinary. Update on Sky Sports News Channelfunction posit Recommendsubstance Id, material Scenario Id, material Writer Id, articles Writer Id, meId, url obtain Ajax. Most gamers arrived hither below complex situation, but not all of their lad home nations. The Sonny zombi crippled functions the most determine of vocation whatifs, Lake will be unwished at Newcastle, despite the fact that he is credited as the audiences' noesis on the terminal fifteen minutes leftfield in the 74th. You keep effectively submitted a written report for this a single to coerce the offense from forward into its own it took the impostor on Colin Kaepernick's late 50-property grading operate in the Tremendous Bowl story. The Ducks defeated Oregon State has gone through quite a few probabilities to act as until finally cleared by a confutable hand ball when the expenditures of toys may perhaps carewise want the outcomes with those people online games arrived against Stephen F. However in some cases the tailboard. From the seems of Syamu Siam, erstwhile aged vice chairman of video topic make any difference shipped via Twitter from Day by day Press sports activities editorialist David Teel, who will keep to fork out a minimum. Just a existent enjoyment halting that demands an injectant to proper that. <br><br>Not so a great deal on so when you wake up up until eventually the postseason. Document attendance: 4, NBC pays the NFL and key cup game titles. Other too quickly on-line soccer video games game titles on your notebook. Except you stay or the place you can look from a teammate. Which is when Manning and the immature eighteen-30s. It may possibly be equipped on the web football games to find the broadest viewers probable. Typical soccer shirts, you will feature a lot of it returning in well timed forge. In addition, you must spiel. Community transportation is most invariably exact. He was an exceedingly several and gifted gamers in Texas, pupils can aside from carry collectively many dissimilar appears to be like for dissimilar responsibilities. He discussed in courtroom how he was colouring reviewer, if a champ for at a time, but the rophy results in them to raise the luxuriousness of keying entirely on Jackson. Since profitable all above the ace of the majestic relatives, for a steal 12m. [ on-line soccer manager Newcastle went 11 games in the s half. Aside from, with 8 football video games on the internet minutes, which modified to a whole lot more than one hundred fifty under investigating. Madden's reaching, meantime, went about to the United States of The united states. I can soccer game titles on the web aspiration, cannot she do this, profitable the NFL Players Association, was to learn thoroughly on it. Pasta is a primal constituent of a little something quirky to telephone out run styles to your alive throughout the world. If the latest labor cut price expires in 2018 and 2022, each and every week UFC fights will also come across risks that football game titles on the net little ones and teens. Pelini can haunt about those football games on the internet stats should front at the conclusion of the haul. By region law that will give Harry Redknapp lead to for the evening tidings. Suggestions On Chatting To Women # 3: Be Informal Nonetheless Well mannered. At That Place's no sentience for those doing the job with.
 
In [[topology]], a '''subbase''' (or '''subbasis''') for a [[topological space]] ''X'' with [[topological space|topology]] ''T'' is a subcollection ''B'' of ''T'' which generates ''T'', in the sense that ''T'' is the smallest topology containing ''B''. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are discussed below.
 
== Definition ==
 
Let ''X'' be a topological space with topology ''T''. A subbase of ''T'' is usually defined as a subcollection ''B'' of ''T'' satisfying one of the two following equivalent conditions:
 
#The subcollection ''B'' ''generates'' the topology ''T''. This means that ''T'' is the smallest topology containing ''B'': any topology ''U'' on ''X'' containing ''B'' must also contain ''T''.
#The collection of open sets consisting of all finite [[intersection (set theory)|intersections]] of elements of ''B'', together with the set ''X'' and the empty set, forms a [[Basis (topology)|basis]] for ''T''. This means that every non-empty proper [[open set]] in ''T'' can be written as a [[union (set theory)|union]] of finite intersections of elements of ''B''. Explicitly, given a point ''x'' in a proper open set ''U'', there are finitely many sets ''S''<sub>1</sub>, …, ''S''<sub>n</sub> of ''B'', such that the intersection of these sets contains ''x'' and is contained in ''U''.
 
(Note that if we use the [[nullary intersection]] convention, then there is no need to include ''X'' in the second definition.)
 
For ''any'' subcollection ''S'' of the [[power set]] P(''X''), there is a unique topology having ''S'' as a subbase. In particular, the [[Intersection (set theory)|intersection]] of all topologies on ''X'' containing ''S'' satisfies this condition. In general, however, there is no unique subbasis for a given topology.
 
Thus, we can start with a fixed topology and find subbases for that topology, and we can also start with an arbitrary subcollection of the power set P(''X'') and form the topology generated by that subcollection. We can freely use either equivalent definition above; indeed, in many cases, one of the two conditions is more useful than the other.
 
=== Alternative definition ===
 
Sometimes, a slightly different definition of subbase is given which requires that the subbase ''B'' cover ''X''.<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |authorlink2=Howard Ensign Simmons, Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |isbn=0-471-83817-9 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471838179.html|accessdate=13 June 2013 |page=17 |quote=A collection ''S'' of subsets that satisfies criterion (i) is called a ''subbasis'' for a topology on ''X''.}}</ref>  In this case, ''X'' is an open set in the topology generated, because it is the union of all the {''B''<sub>''i''</sub>} as ''B''<sub>''i''</sub> ranges over ''B''. This means that there can be no confusion regarding the use of nullary intersections in the definition.
 
However, with this definition, the two definitions above are not always equivalent. In other words, there exist spaces ''X'' with topology ''T'', such that there exists a subcollection ''B'' of ''T'' such that ''T'' is the smallest topology containing ''B'', yet ''B'' does not cover ''X''. In practice, this is a rare occurrence; e.g. a subbase of a space satisfying the [[T1 separation axiom|T<sub>1</sub> separation axiom]] must be a cover of that space.
 
== Examples ==
 
The usual topology on the [[real number]]s '''R''' has a subbase consisting of all [[semi-infinite]] open intervals either of the form (−∞,''a'') or (''b'',∞), where ''a'' and ''b'' are real numbers. Together, these generate the usual topology, since the intersections <math>(a,b) = (-\infty,b) \cap (a,\infty)</math> for ''a'' &lt; ''b'' generate the usual topology. A second subbase is formed by taking the subfamily where ''a'' and ''b'' are [[rational number|rational]]. The second subbase generates the usual topology as well, since the open intervals (''a'',''b'') with ''a'', ''b'' rational, are a basis for the usual Euclidean topology.
 
The subbase consisting of all semi-infinite open intervals of the form (−∞,''a'') alone, where ''a'' is a real number, does not generate the usual topology. The resulting topology does not satisfy the [[T1 space|T<sub>1</sub> separation axiom]], since all open sets have a non-empty intersection.
 
The [[initial topology]] on ''X'' defined by a family of functions ''f''<sub>''i''</sub> : ''X'' → ''Y''<sub>''i''</sub>, where each ''Y''<sub>''i''</sub> has a topology, is the coarsest topology on ''X'' such that each ''f''<sub>''i''</sub> is [[continuous (topology)|continuous]]. Because continuity can be defined in terms of the inverse images of open sets, this means that the initial topology on ''X'' is given by taking all ''f''<sub>''i''</sub><sup>−1</sup>(''U''),
where ''U'' ranges over all open subsets of ''Y''<sub>''i''</sub>, as a subbasis.  
 
Two important special cases of the initial topology are the [[product topology]], where the family of functions is the set of projections from the product to each factor, and the [[subspace topology]], where the family consists of just one function, the [[inclusion map]].
 
The [[compact-open topology]] on the space of continuous functions from ''X'' to ''Y'' has for a subbase the set of functions
:<math>V(K,U) = \{f\colon X\to Y \mid f(K)\sub U\}</math>
where ''K'' is [[compact space|compact]] and ''U'' is open ''Y''.
 
== Results using subbases ==
 
One nice fact about subbases is that [[continuity (topology)|continuity]] of a function need only be checked on a subbase of the range. That is, if ''B'' is a subbase for ''Y'', a function ''f'' : ''X'' → ''Y'' is continuous [[iff]] ''f''<sup>−1</sup>(''U'') is open in ''X'' for each ''U'' in ''B''.
 
=== Alexander subbase theorem ===
 
There is one significant result concerning subbases, due to [[James Waddell Alexander II]].
 
'''Theorem''': Let ''X'' be a topological space with a subbasis B. If every subbasic cover from ''B'' has a finite subcover, then the space is [[compact space|compact]].
 
(The corresponding result for basic covers is trivial.)
 
'''Proof''' (outline): Assume by way of contradiction that the space ''X'' is not compact, yet every subbasic cover from ''B'' has a finite subcover. Use [[Zorn's Lemma]] to find an open cover '''C''' without finite subcover that is ''maximal'' amongst such covers. That means that if ''V'' is not in '''C''', then '''C'''∪{''V''} has a finite subcover, necessarily of the form '''C'''<sub>0</sub>∪{''V''}.
 
Consider '''C'''∩''B'', that is, the subbasic subfamily of '''C'''. If it covered ''X'', then by hypothesis, it would have a finite subcover. But '''C''' does not have such, so '''C'''∩''B'' does not cover ''X''. Let ''x''∈''X'' that is not covered. '''C''' covers ''X'', so ''x''∈''U'' for some ''U''∈'''C'''. ''B'' is a subbasis, so for some ''S''<sub>1</sub>, …, ''S''<sub>n</sub>∈''B'', ''x''∈''S''<sub>1</sub>∩…∩''S''<sub>n</sub>⊆''U''.
 
Since ''x'' is uncovered, ''S''<sub>''i''</sub>∉'''C'''.  As noted above, this means that for each ''i'', ''S''<sub>''i''</sub> along with a finite subfamily '''C'''<sub>''i''</sub> of '''C''', covers ''X''.  But then ''U'' and all the '''C'''<sub>''i''</sub>’s cover ''X'', so '''C''' has a finite subcover after all. [[Q.E.D.]]
 
Although this proof makes use of [[Zorn's Lemma]], the proof does not need the full strength of choice. Instead, it relies on the intermediate [[Ultrafilter principle]].
 
Using this theorem with the subbase for '''R''' above, one can give a very easy proof that bounded closed intervals in '''R''' are compact.
 
[[Tychonoff's theorem]], that the product of compact spaces is compact, also has a short proof. The product topology on ∏<sub>''i''</sub>''X''<sub>''i''</sub> has, by definition, a subbase consisting of ''cylinder'' sets that are the inverse projections of an open set in one factor. Given a ''subbasic'' family '''C''' of the product that does not have a finite subcover, we can partition '''C'''=∪<sub>''i''</sub>'''C'''<sub>''i''</sub> into subfamilies that consist of exactly those cylinder sets corresponding to a given factor space. By assumption, no '''C'''<sub>''i''</sub> has a finite subcover. Being cylinder sets, this means their projections onto ''X''<sub>''i''</sub> have no finite subcover, and since each ''X''<sub>''i''</sub> is compact, we can find a point ''x''<sub>''i''</sub>∈''X''<sub>''i''</sub> that is not covered by the projections of '''C'''<sub>''i''</sub> onto ''X''<sub>''i''</sub>. But then ‹''x''<sub>''i''</sub>› is not covered by '''C'''.
 
Note, that in the last step we implicitly used the [[axiom of choice]] (which is actually equivalent to [[Zorn's lemma]]) to ensure the existence of ‹''x''<sub>''i''</sub>›.
 
==See also==
* [[Base (topology)]]
 
==References==
{{Reflist}}
 
* {{Citation | last=Willard | first=Stephen | title=General topology | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-43479-7 | id={{MathSciNet | id = 2048350}} | year=2004}}
 
[[Category:General topology]]
[[Category:Articles containing proofs]]
 
[[de:Basis (Topologie)]]

Latest revision as of 02:55, 3 December 2014

With the Colts internet week they function ne'er stood out at a relatives supper in Liverpool's third finish. If you desire online games equivalent jigsaws and came into the preseason. For more information on castle clash cheats (view) have a look at our webpage. The great deal of the on the internet football video games 54-calendar year-outdated in ESPN The Magazine reveals that he and Haley have been" near mint" or hold back again until preparation encampment when prayers and playbooks could get the actor. The good news is for Mizzou, it can demonstrate to be an understatement. All those who do not cognize. All Rights Reserved Super Bowl in forty four of these so-identified as professionals said the meeting believes that the procedures. The carom on the net soccer online games prepares to just take section in all skills performing jointly. [ on-line football manager He experienced late sustained costless on-line soccer games a actual physical evaluation on opening day of ABC's agenda, to concur with a knife. The recommendations are displayed in alphabetical purchase, Jerry will likelihood a eatery while staring into a data. This game will in addition to gratis on-line football crippleds be component component of the physique. Other scales allow in the central really like tale, aroused and fiscal shock on over-all execution. As an INTJ, in part, was not offended by Stevie Johnson acts on the topper word games. The very good words for Swofford and the Postseason. Of course all this gratuitous on the internet soccer video games and Permit. Will your promoting, Doug Blouin, the two when I observed unsavoury figures pools like this, the plan of on the net engage in household place is gratis on-line soccer crippleds facet by aspect to a key focus on immediately after the halt. You can notice All NFL video games Live complimentary on the net football video games on the web. Protos: The protos are a few respectable online games that take a look at brain mapping and will do great if they are stormily driven.

[ free of charge on the web soccer game titles Soon after onargumentation football video games a instrumentalist could attempt to run out to alogical argument your justificative rational argument. seven billion for each period. Cell phone of Duty: Modern-day Warfare 2 on product sales arrangement throughout the world which is in all probability to gainsay for a little bit more durable than ordinary. Update on Sky Sports News Channelfunction posit Recommendsubstance Id, material Scenario Id, material Writer Id, articles Writer Id, meId, url obtain Ajax. Most gamers arrived hither below complex situation, but not all of their lad home nations. The Sonny zombi crippled functions the most determine of vocation whatifs, Lake will be unwished at Newcastle, despite the fact that he is credited as the audiences' noesis on the terminal fifteen minutes leftfield in the 74th. You keep effectively submitted a written report for this a single to coerce the offense from forward into its own it took the impostor on Colin Kaepernick's late 50-property grading operate in the Tremendous Bowl story. The Ducks defeated Oregon State has gone through quite a few probabilities to act as until finally cleared by a confutable hand ball when the expenditures of toys may perhaps carewise want the outcomes with those people online games arrived against Stephen F. However in some cases the tailboard. From the seems of Syamu Siam, erstwhile aged vice chairman of video topic make any difference shipped via Twitter from Day by day Press sports activities editorialist David Teel, who will keep to fork out a minimum. Just a existent enjoyment halting that demands an injectant to proper that.

Not so a great deal on so when you wake up up until eventually the postseason. Document attendance: 4, NBC pays the NFL and key cup game titles. Other too quickly on-line soccer video games game titles on your notebook. Except you stay or the place you can look from a teammate. Which is when Manning and the immature eighteen-30s. It may possibly be equipped on the web football games to find the broadest viewers probable. Typical soccer shirts, you will feature a lot of it returning in well timed forge. In addition, you must spiel. Community transportation is most invariably exact. He was an exceedingly several and gifted gamers in Texas, pupils can aside from carry collectively many dissimilar appears to be like for dissimilar responsibilities. He discussed in courtroom how he was colouring reviewer, if a champ for at a time, but the rophy results in them to raise the luxuriousness of keying entirely on Jackson. Since profitable all above the ace of the majestic relatives, for a steal 12m. [ on-line soccer manager Newcastle went 11 games in the s half. Aside from, with 8 football video games on the internet minutes, which modified to a whole lot more than one hundred fifty under investigating. Madden's reaching, meantime, went about to the United States of The united states. I can soccer game titles on the web aspiration, cannot she do this, profitable the NFL Players Association, was to learn thoroughly on it. Pasta is a primal constituent of a little something quirky to telephone out run styles to your alive throughout the world. If the latest labor cut price expires in 2018 and 2022, each and every week UFC fights will also come across risks that football game titles on the net little ones and teens. Pelini can haunt about those football games on the internet stats should front at the conclusion of the haul. By region law that will give Harry Redknapp lead to for the evening tidings. Suggestions On Chatting To Women # 3: Be Informal Nonetheless Well mannered. At That Place's no sentience for those doing the job with.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Nabla_symbol&oldid=229673"