Editing Topologist's sine curve (section)

==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]].