Neighbourhood (mathematics): Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
en>Marc van Woerkom
en>Mark viking
→‎top: you -> one
Line 1: Line 1:
In [[mathematics]], particularly [[topology]], the '''homeomorphism group''' of a [[topological space]] is the [[group (mathematics)|group]] consisting of all [[homeomorphism]]s from the space to itself with [[function composition]] as the group [[binary operation|operation]]. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of [[automorphism group]]s. Homeomorphism groups are [[topological invariant]]s in the sense that the homeomorphism groups of homeomorphic topological spaces are [[Group isomorphism|isomorphic as groups]].
I woke up yesterday  and realised - Now I have also been  [http://lukebryantickets.neodga.com buy luke bryan tickets] solitary for some time and after much bullying from friends I now find myself registered for internet dating. They assured me that there are a lot of sweet, standard and interesting individuals to meet up, so here goes the pitch!<br>My family and buddies are wonderful and spending time  [http://minioasis.com luke bryan winter tour] together at bar gigs or dishes is definitely critical. As I come to realize you could not get a nice conversation against the noise I have never been into night clubs. In addition, I have two really adorable and unquestionably cheeky canines who are invariably enthusiastic to meet up [http://Search.Huffingtonpost.com/search?q=fresh+people&s_it=header_form_v1 fresh people].<br>I make an effort to stay as physically fit as possible coming to the gym several-times a week. I appreciate my athletics and try to perform or view as several a possible. I am going to often at Hawthorn matches being winter. Notice: I've seen the carnage of fumbling fits at stocktake revenue, Supposing that you would considered purchasing a sport I do not brain.<br><br>Take a look at my weblog: [http://lukebryantickets.lazintechnologies.com luke bryan tour lineup]
 
==Properties and Examples==
There is a natural [[group action]] of the homeomorphism group of a space on that space. If this action is [[transitive group action|transitive]], then the space is said to be [[Homogeneous space|homogeneous]].
 
===Topology===
{{Expand section|date=March 2009}}
As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the [[compact-open topology]] (in the case of regular, locally compact spaces), making it into a [[topological group]].
 
In the category of topological spaces with homeomorphisms, object groups are exactly homeomorphism groups.
 
==Mapping class group==
{{Main|Mapping class group}}
In [[geometric topology]] especially, one considers the [[quotient group]] obtained by quotienting out by [[Homotopy#Isotopy|isotopy]], called the [[mapping class group]]:
:<math>{\rm MCG}(X) = {\rm Homeo}(X) / {\rm Homeo}_0(X)</math>
The MCG can also be interpreted as the 0th [[homotopy group]], <math>{\rm MCG}(X) = \pi_0({\rm Homeo}(X))</math>.
This yields the [[short exact sequence]]:
:<math>1 \rightarrow {\rm Homeo}_0(X) \rightarrow {\rm Homeo}(X) \rightarrow {\rm MCG}(X) \rightarrow 1.</math>
 
In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.
 
==See also==
* [[Mapping class group]]
 
==References==
*{{Springer|id=H/h047610|title=homeomorphism group}}
 
{{DEFAULTSORT:Homeomorphism Group}}
[[Category:Group theory]]
[[Category:Topology]]
[[Category:Topological groups]]

Revision as of 05:52, 21 February 2014

I woke up yesterday and realised - Now I have also been buy luke bryan tickets solitary for some time and after much bullying from friends I now find myself registered for internet dating. They assured me that there are a lot of sweet, standard and interesting individuals to meet up, so here goes the pitch!
My family and buddies are wonderful and spending time luke bryan winter tour together at bar gigs or dishes is definitely critical. As I come to realize you could not get a nice conversation against the noise I have never been into night clubs. In addition, I have two really adorable and unquestionably cheeky canines who are invariably enthusiastic to meet up fresh people.
I make an effort to stay as physically fit as possible coming to the gym several-times a week. I appreciate my athletics and try to perform or view as several a possible. I am going to often at Hawthorn matches being winter. Notice: I've seen the carnage of fumbling fits at stocktake revenue, Supposing that you would considered purchasing a sport I do not brain.

Take a look at my weblog: luke bryan tour lineup

Retrieved from "https://en.formulasearchengine.com/index.php?title=Neighbourhood_(mathematics)&oldid=238754"