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In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example.
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It can be defined as the [[graph of a function|graph]] of the function sin(1/''x'') on the half-open interval (0,&nbsp;1], together with the origin, under the topology [[subspace topology|induced]] from the [[Euclidean plane]]:
 
:<math> T = \left\{  \left( x, \sin \frac{1}{x}  \right ) :  x \in (0,1] \right\} \cup \{(0,0)\}. </math>
 
==Image of the curve==
[[Image:Topologist's sine curve.svg|420px|Topologist's Sine Curve]]
 
As ''x'' approaches zero from the right, the magnitude of the rate of change of 1/''x'' increases. This is why the frequency of the sine wave increases as one moves to the left in the graph.
 
==Properties==
The topologist's sine curve ''T'' is [[connected space|connected]] but neither [[locally connected space|locally connected]] nor [[connected space#Path connectedness|path connected]]. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a [[path (topology)|path]].
 
The space ''T'' is the continuous image of a [[locally compact]] space (namely, let ''V'' be the space {&minus;1} &cup; (0,&nbsp;1<nowiki>]</nowiki>, and use the map ''f'' from ''V'' to ''T'' defined by <span style="white-space: nowrap">''f''(&minus;1)</span> = (0,0) and <span style="white-space: nowrap">''f''(''x'')</span> = <span style="white-space: nowrap">(''x'',&nbsp;sin(1/''x''))</span> for ''x'' > 0), but ''T'' is not locally compact itself.
 
The [[topological dimension]] of ''T'' is 1.
 
==Variants==
Two variants of the topologist's sine curve have other interesting properties.
 
The '''closed topologist's sine curve''' can be defined by taking the topologist's sine curve and adding its set of [[limit point]]s, <math>\{(0,y)\mid y\in[-1,1]\}</math>. This space is closed and bounded and so [[compact space|compact]] by the [[Heine–Borel theorem]], but has similar properties to the topologist's sine curve&mdash;it too is connected but neither locally connected nor path-connected.
 
The '''extended topologist's sine curve''' can be defined by taking the closed topologist's sine curve and adding to it the set <math>\{(x,1) \mid x\in[0,1]\}</math>. It is [[arc connected]] but not [[Locally connected space|locally connected]].
 
==References==
*{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=Dover Publications, Inc. | location=Mineola, NY | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | id={{MathSciNet|id=1382863}} | year=1995 | pages=137–138}}
*{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}}
 
[[Category:Topological spaces]]

Latest revision as of 20:28, 9 December 2014

From comprehending the wedding party specialist you should consult to identifying ovulation with 100% precision, means that inside the book. To fall pregnant everything must be in the right place so be patient and keep trying. You will need to take into consideration always currently being beneficial. This article is based on the book, "Pregnancy Miracle" by Lisa Olson who dedicated her life to creating the ultimate pregnancy solution to permanently reverse the root of infertility and dramatically improve the overall quality of your life, without the use prescription medication and without any surgical procedures. after getting an avalanche of complaints from viewers, who said she was a horrible and unprofessional host.

Coffee, soda, and chocolate can reduce the quality of sperm and egg cells. It may interferes with the embryo implantation if the uterine tissue extends through entire region of the uterine wall resulting in miscarriage. Backache can occur for a number of reasons, which can be reduced to two main issues: hormones and postural changes. Woman ovulate about the 14th day so having intercourse during that time is crucial if you want to conceive. - You should try to stay away an evil magic spell to get pregnant that can lower your chances of getting pregnant, such as, alcohol, cigarettes, and any forms of caffeine.

There are ovulation prediction kits and saliva tests available and you can get them over the counter at your local pharmacy. For successful attempts at conception, your natural flow of energy must be in balance through full function of your kidneys, liver and spleen. Pregnancy is meant to be one of the happiest times of a woman's life. There will be boost in dimensions as couples age and the woman's monthly ovulation decreases. In this case the egg is normal, but for what ever reason, the sperm can not penetrate it.

Normally, it takes several months or sometimes even years to get pregnant. Lisa Olson, a certified nutritionist, health consultant and author has not just pumped out yet another "pregnancy program" into an already over-saturated market. ” This guide on The Sims 3 Woo - Hoo will help to explain how you can get your Sims to Woo - Hoo, and where Woo - Hoo is able to happen. The luteal phase and begins the day of ovulation and ends on the day you begin to menstruate. Another trick to help you get pregnant quick is to lie still with your pelvic area lifted by a pillow for at least twenty minutes after intercourse.

The third and doubtless most correct technique to know when you will ovulate is to make use of an ovulation predictor kit. Specialists exclaim that it's innocent for women to have up to 12 ounces of low mercury fish per week. Also higher than normal levels of stress hormones can lower libido in both men and women because they have a knock on effect on the hormones that power sex drive - oestrogen and testosterone. Aside from a high quality fertility supplement, you should include your diet with fertility super foods to increase the rate of getting pregnant. Since every female is different from one another, this includes her hormone balance which in turn affects her period cycles, there is no exact science saying when a woman is fertile and when one is not.

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