Topologist's sine curve

Apr 22, 2026 - 20:04
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Topologist's sine curve

link author: James Munkres (via WP:JWB)

← Previous revision Revision as of 14:33, 22 April 2026
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: T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}.
: T = \left\{ \left( x, \sin \tfrac{1}{x} \right ) : x \in (0,1] \right\} \cup \{(0,0)\}.




==Properties==
==Properties==
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Two variants of the topologist's sine curve have other interesting properties:
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, \{(0,y)\mid y\in[-1,1]\}; 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.{{Cite book |last=Munkres |first=James R |title=Topology; a First Course |date=1979 |publisher=Englewood Cliffs |isbn=9780139254956 |page=158}} 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 ''closed'' topologist's sine curve can be defined by taking the topologist's sine curve and adding its set of [[limit point]]s, \{(0,y)\mid y\in[-1,1]\}; 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.{{Cite book |last=Munkres |first=James R |author-link=James Munkres |title=Topology; a First Course |date=1979 |publisher=Englewood Cliffs |isbn=9780139254956 |page=158}} 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 \{(x,1) \mid x\in[0,1]\}. This variant is [[arc connected]] but not [[Locally connected space|locally connected]].
*The ''extended'' topologist's sine curve can be defined by taking the closed topologist's sine curve and adding to it the set \{(x,1) \mid x\in[0,1]\}. This variant is [[arc connected]] but not [[Locally connected space|locally connected]].
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