Topologist's sine curve: Difference between revisions
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In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' or '''Warsaw sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example. | In the branch of [[mathematics]] known as [[topology]], the '''topologist's sine curve''' or '''Warsaw sine curve''' is a [[topological space]] with several interesting properties that make it an important textbook example. | ||
It can be defined as the [[graph of a function|graph]] of the function sin(1 | It can be defined as the [[graph of a function|graph]] of the function <math> \sin \big( \frac {1}{x} \big) </math> on the [[half-open interval]] <math> ( 0 , 1 ] </math>, together with the origin, under the topology [[subspace topology|induced]] from the [[Euclidean plane]]: | ||
:<math> T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math> | :<math> T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}. </math> | ||
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==Variants== | ==Variants== | ||
Two variants of the topologist's sine curve have other interesting properties: | |||
The ''' | *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>; some texts define the topologist's sine curve itself as this closed version, as they prefer to use the term 'closed topologist's sine curve' to refer to another curve.<ref>{{Cite book |last=Munkres |first=James R |title=Topology; a First Course |date=1979 |publisher=Englewood Cliffs |isbn=9780139254956 |page=158}}</ref> 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. | ||
*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>. This variant is [[arc connected]] but not [[Locally connected space|locally connected]]. | |||
== See also == | == See also == | ||
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==References== | ==References== | ||
{{reflist}} | {{reflist}} | ||
*{{Citation | | *{{Citation |last=Steen |first=Lynn Arthur |title=[[Counterexamples in Topology]] |pages=137–138 |year=1995 |orig-date=1978 |edition=[[Dover Publications|Dover]] reprint of 1978 |place=Mineola, NY |publisher=Dover Publications, Inc. |isbn=978-0-486-68735-3 |mr=1382863 |last2=Seebach |first2=J. Arthur Jr. |author-link=Lynn Arthur Steen |author-link2=J. Arthur Seebach, Jr.}} | ||
*{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}} | *{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}} | ||
[[Category:Topological spaces]] | [[Category:Topological spaces]] | ||
Latest revision as of 11:12, 5 June 2025
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the function on the half-open interval , together with the origin, under the topology induced from the Euclidean plane:
Properties
The topologist's sine curve Template:Mvar is connected but neither locally connected nor path connected. This is because it includes the point Template:Math but there is no way to link the function to the origin so as to make a path.
The space Template:Mvar is the continuous image of a locally compact space (namely, let Template:Mvar be the space and use the map defined by and for Template:Math), but Template:Mvar is not locally compact itself.
The topological dimension of Template:Mvar 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 points, ; some texts define the topologist's sine curve itself as this closed version, as they prefer to use the term 'closed topologist's sine curve' to refer to another curve.[1] This space is closed and bounded and so 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.
- The extended topologist's sine curve can be defined by taking the closed topologist's sine curve and adding to it the set . This variant is arc connected but not locally connected.
See also
References
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