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In [[mathematics]], the '''Hawaiian earring''' ''H'' is  the [[topological space]] defined by the [[union (set theory)|union]] of circles in the [[Euclidean plane]] '''R'''<sup>2</sup> with center (1/''n'',&nbsp;0) and radius 1/''n'' for ''n''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;.... The space ''H'' is [[homeomorphism|homeomorphic]] to the [[one-point compactification]] of the union of a countably infinite family of [[interval (mathematics)|open intervals]].
[[Image:Hawaiian Earrings.svg|right|250px|thumb|The Hawaiian earring. Only the ten largest circles are shown.]]


The Hawaiian earring can be given a [[complete space|complete metric]] and it is [[compact space|compact]].  It is [[path connected]] but not [[semilocally simply connected]].


The Hawaiian earring looks very similar to the [[wedge sum]] of countably infinitely many circles; that is, the [[rose (topology)|rose]] with infinitely many petals, but those two spaces are not homeomorphic. The difference between their topologies is seen in the fact that, in the Hawaiian earring, every open neighborhood of the point of intersection of the circles contains all but finitely many of the circles. It is also seen in the fact that the wedge sum is not compact: the complement of the distinguished point is a union of open intervals; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.
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== Fundamental group ==
 
The Hawaiian earring is not simply connected, since the loop parametrising any circle is not homotopic to a trivial loop. Thus, it has a nontrivial [[fundamental group]]&nbsp;''G''.
 
The Hawaiian earring ''H'' has the [[free group]] of countably infinitely many generators as a proper subgroup of its [[fundamental group]]. ''G''&nbsp;contains additional elements which arise from loops whose image is not contained in finitely many of the Hawaiian's earrings circles;  in fact, some of them are surjective. For example, the path that on the interval [2<sup>&minus;''n''</sup>,&nbsp;2<sup>&minus;(''n''&minus;1)</sup>] circumnavigates the ''n''th circle.
 
It has been shown that ''G'' [[embedding|embeds]] into the [[inverse limit]] of the free groups with ''n'' generators, ''F''<sub>''n''</sub>, where the bonding map from ''F''<sub>''n''</sub> to ''F''<sub>''n''&minus;1</sub> simply kills the last generator of&nbsp;''F''<sub>''n''</sub>.  However ''G'' is not the complete inverse limit but rather the subgroup in which each generator appears only finitely many times. An example of an element of the inverse limit which is not an element of ''G'' is an infinite commutator.
 
''G'' is uncountable, and it is not a free group. While its [[abelianisation]] has no known simple description, ''G'' has a normal subgroup ''N'' such that <math>\textstyle G/N \approx \prod_{i=0}^\infty \mathbb{Z}</math>, the [[direct product of groups|direct product]] of infinitely many copies of the [[infinite cyclic group]] (the [[Baer–Specker group]]).  This is called the ''infinite abelianization'' or ''strong abelianization'' of the Hawaiian earring, since the subgroup ''N'' is generated by elements where each coordinate (thinking of the Hawaiian earring as a subgroup of the inverse limit) is a product of commutators. In a sense, ''N'' can be thought of as the closure of the commutator subgroup.
 
==References==
*{{citation|first1=J. W.|last1=Cannon|first2=G. R.|last2=Conner|title=The big fundamental group, big Hawaiian earrings, and the big free groups|journal=Topology and its Applications|volume=106|year=2000|issue=3|pages=273–291|doi=10.1016/S0166-8641(99)00104-2}}.
*{{citation|first1=G.|last1=Conner|first2=K.|last2=Spencer|title=Anomalous behavior of the Hawaiian earring group|journal=Journal of Group Theory|volume=8|year=2005|issue=2|pages=223–227|doi=10.1515/jgth.2005.8.2.223}}.  
*{{citation|authorlink=Katsuya Eda|first=K.|last=Eda|title=The fundamental groups of one-dimensional wild spaces and the Hawaiian earring|url=http://www.ams.org/journals/proc/2002-130-05/S0002-9939-01-06431-0/S0002-9939-01-06431-0.pdf|format=PDF|journal=Proceedings of the American Mathematical Society|volume=130|year=2002|issue=5|pages=1515–1522|doi=10.1090/S0002-9939-01-06431-0}}.
*{{citation|first1=K.|last1=Eda|first2=K.|last2=Kawamura|title=The singular homology of the Hawaiian earring|journal=Journal of the London Mathematical Society|volume=62|year=2000|issue=1|pages=305–310|doi=10.1112/S0024610700001071}}.  
*{{citation|first=P.|last=Fabel|title=The topological Hawaiian earring group does not embed in the inverse limit of free groups|url=http://arxiv.org/pdf/math/0501482.pdf|format=PDF|journal=Algebraic & Geometric Topology|volume=5|year=2005|pages=1585–1587|doi=10.2140/agt.2005.5.1585}}.
*{{citation|author1-link=John Morgan (mathematician)|first1=J. W.|last1=Morgan|first2=I.|last2=Morrison|title=A van Kampen theorem for weak joins|journal=Proceedings of the London Mathematical Society|volume=53|issue=3|year=1986|pages=562–576|doi=10.1112/plms/s3-53.3.562}}.
*{{citation|first=Daniel K.|last=Biss|authorlink=Daniel Biss|title=A generalized approach to the fundamental group|url=http://math.mit.edu/~gracelyo/18904/BissCoveringSpaceArticle.pdf|format=PDF|journal=[[American Mathematical Monthly]]|volume=107|issue=8|year=2000|pages=711–720|doi=10.2307/2695468|publisher=Mathematical Association of America|jstor=2695468}}.
 
[[Category:Topological spaces]]

Latest revision as of 13:20, 8 April 2014


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While we work from home, things can end up gobbling up that irreplaceable time we need to produce. If we fall prey to such distractions while we work from home, the distractions we expect to last only a few minutes just might turn into several hours of costly down time. It is so easy to be distracted by low-priority things when we work from home. Those few idle minutes between phone calls or E-mails can entice us from an awaiting project to finish taking out the garbage, or put that remaining load of clothes into the dryer. Of course we understand it's serious that we work from home - we know our work is important but, while we work from home, the grocery list needs updating; although we work from home, we still have that recorded game to watch.

For starters, the internet has helped everyone see things that could have only been possible if they travelled. To date, the most convenient that has been done through the World Wide Web is how people do business which is by E-commerce. The internet has made the home into the virtual substitute of the physical store, other than that, it has also become the office of many employed people all over the world. The last 15 years have been called the Internet Era and indeed it has made many contributions on how everyone has lived their lives.

It has made going to work easy and breezy as you don't have to get dressed- a computer connection and computer is all you need. Many jobs are available online such as data entry, programming, graphic design and even medical transcription. In a typical scenario, the online employee logs on to his computer, speaks to the client and does or finishes the task for the day- all in his pajamas at home. Anyone can work at home as there are many websites that post hundreds of classification ads looking for different computer literate individuals.

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Being our own boss while we work from home is an especially serious endeavor; both our current and future livelihoods depend on our keeping our professional selves in line. We must oversee how we actually spend our time while we work from home, in order to ensure that we do the very best job we can. Our timely completions of well-done jobs while we work from home are noticed by those upon whom our livelihoods depend.

Go to your upline and find a mentor. Make sure you have all of the necessary tools to ensure that you can drive traffic to your website and build your contact list . Those who do not do this, soon have their work from home dream turn into a nightmare. Once you have the perfect work from home opportunity. The best way to do this is educate yourself and stay on the cutting edge on what�s new. Just remember that only a fool knows everything. The people that do this on a regular basis are the ones that win.

On the other hand, many people don"t have the awareness or ability set to have achievement in this challenging but rewarding business. Given the present situation of our financial system and the job market there are a lot of individuals searching for work from home employment opportunities. Deciding to work from home is a simple choice for most individuals.

The key to making work from home work for you is promotion. Figure out who is most likely to buy your product or pay for your services, and target them with marketing and promotional offers. No matter what business you choose, you need to get the word out, and get it out to the right markets. The best business idea in the world will fall flat if no one knows about it. Depending on what you offer, that may mean spreading your flyers around the neighborhood, taking out newspaper or radio ads, sending out email newsletters, posting on bulletin boards and other Internet services, asking for referrals or any combination of the above.

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