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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, 1], together with the origin, under the topology [[subspace topology|induced]] from the [[Euclidean plane]]:
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| :<math> T = \left\{ \left( x, \sin \frac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math>
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| ==Image of the curve==
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| [[Image:Topologist's sine curve.svg|420px|Topologist's Sine Curve]]
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| 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.
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| ==Properties==
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| 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]].
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| The space ''T'' is the continuous image of a [[locally compact]] space (namely, let ''V'' be the space {−1} ∪ (0, 1<nowiki>]</nowiki>, and use the map ''f'' from ''V'' to ''T'' defined by <span style="white-space: nowrap">''f''(−1)</span> = (0,0) and <span style="white-space: nowrap">''f''(''x'')</span> = <span style="white-space: nowrap">(''x'', sin(1/''x''))</span> for ''x'' > 0), but ''T'' is not locally compact itself. | |
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| The [[topological dimension]] of ''T'' is 1.
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| ==Variants== | |
| Two variants of the topologist's sine curve have other interesting properties.
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| 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—it too is connected but neither locally connected nor path-connected.
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| 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]].
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| ==References==
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| *{{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}}
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| *{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}}
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| [[Category:Topological spaces]]
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