20:
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390:; 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. This space is closed and bounded and so
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398:, 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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increases. This is why the frequency of the sine wave increases as one moves to the left in the graph.
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192:{\displaystyle T=\left\{\left(x,\sin {\tfrac {1}{x}}\right):x\in (0,1]\right\}\cup \{(0,0)\}.}
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can be defined by taking the closed topologist's sine curve and adding to it the set
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reprint of 1978 ed.), Mineola, NY: Dover
Publications, Inc., pp. 137–138,
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55:
with several interesting properties that make it an important textbook example.
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Two variants of the topologist's sine curve have other interesting properties.
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can be defined by taking the topologist's sine curve and adding its set of
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approaches zero from the right, the magnitude of the rate of change of 1/
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be the space {−1} ∪ (0, 1], and use the map
70:(0, 1], together with the origin, under the topology
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460:{\displaystyle \{(x,1)\mid x\in \}}
383:{\displaystyle \{(0,y)\mid y\in \}}
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513:. Englewood Cliffs. p. 158.
403:extended topologist's sine curve
295:is not locally compact itself.
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319:closed topologist's sine curve
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16:Pathological topological space
1:
237:is the continuous image of a
210:The topologist's sine curve
543:Counterexamples in Topology
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577:"Topologist's Sine Curve"
509:Munkres, James R (1979).
58:It can be defined as the
511:Topology; a First Course
45:topologist's sine curve
538:Seebach, J. Arthur Jr.
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62:of the function sin(1/
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300:topological dimension
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396:Heine–Borel theorem
241:space (namely, let
601:Topological spaces
574:Weisstein, Eric W.
534:Steen, Lynn Arthur
485:List of topologies
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380:
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68:half-open interval
33:
557:978-0-486-68735-3
473:locally connected
220:locally connected
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53:topological space
49:Warsaw sine curve
35:In the branch of
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76:Euclidean plane
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291:> 0), but
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323:limit points
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264:= (0,0) and
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218:but neither
211:
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257:defined by
37:mathematics
496:References
262:(−1)
233:The space
206:Properties
582:MathWorld
540:(1995) ,
437:∈
431:∣
363:−
357:∈
351:∣
216:connected
163:∪
140:∈
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74:from the
66:) on the
39:known as
595:Category
479:See also
471:but not
467:. It is
310:Variants
41:topology
566:1382863
394:by the
392:compact
72:induced
564:
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306:is 1.
43:, the
548:Dover
249:from
60:graph
51:is a
552:ISBN
515:ISBN
401:The
317:The
298:The
287:for
228:path
222:nor
302:of
253:to
214:is
111:sin
47:or
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
