Editing Topologist's sine curve

{{Short description|Pathological topological space}}
[[Image:Topologist's sine curve.svg|420px|thumb| 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.]]

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/''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 \tfrac{1}{x}  \right ) :  x \in (0,1] \right\} \cup \{(0,0)\}. </math>



==Properties==

The topologist's sine curve {{mvar|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 {{math|(0, 0)}} but there is no way to link the function to the origin so as to make a [[path (topology)|path]].

The space {{mvar|T}} is the continuous image of a [[locally compact]] space (namely, let {{mvar|V}} be the space <math>\{-1\} \cup (0, 1],</math> and use the map <math>f : V \to T</math> defined by <math>f(-1) = (0,0)</math> and <math>f(x) = (x, \sin\tfrac{1}{x})</math> for {{math|''x'' > 0}}), but {{mvar|T}} is not locally compact itself.

The [[topological dimension]] of {{mvar|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>; 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 |date=1979 |title=Topology; a First Course |publisher=Englewood Cliffs |page=158 |isbn=9780139254956}}</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&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]].

== See also ==

* [[List of topologies]]
* [[Warsaw circle]]

==References==
{{reflist}}
*{{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]] | orig-date=1978 | publisher=Dover Publications, Inc. | location=Mineola, NY | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 |mr=1382863 | year=1995 | pages=137–138}}
*{{mathworld|urlname=TopologistsSineCurve|title=Topologist's Sine Curve}}

[[Category:Topological spaces]]