Talk:Closed Topologist's Sine Curve is not Path-Connected

Rename?
Regarding Dfeuer's suggestion of renaming, Topologist's Sine Curve is also the name I've learned for this example. But technically it's not the graph $G$ that is adressed here, but the closure $G \cup J$ of the graph. So if we should rename, I'd suggest Topologist's Sine Curve is not Path-Connected.

Also, if the theorem is renamed, there is an accompanying theorem Graph of Sine of Reciprocal is Connected that should be moved as well. --Anghel (talk) 16:28, 12 January 2013 (UTC)


 * I believe that the topologist's sine curve is that closure. --Dfeuer (talk) 16:34, 12 January 2013 (UTC)


 * Steen and Seebach use "Topologist's Sine Curve" for $G \cup \{(0,0)\}$, the closed topologist's sine curve is the one we need here, and then the "extended topologist's sine curve" is the closed one along with the segment from $(0,1)$ to $(1,1)$. Although I don't like S&amp;S as a source for general topological terminology, I think they're probably definitive on counterexample nomenclature. --Dfeuer (talk) 16:44, 12 January 2013 (UTC)


 * I just created Definition:Topologist's Sine Curve. It's a bit of a mess, so feel free to try to make it better. --Dfeuer (talk) 16:50, 12 January 2013 (UTC)


 * Wikipedia agrees with S&S, while PlanetMath and Munkres' Topology (if memory serves me right) use "Topologist's Sine Curve" for $G \cup J$. So we can choose our own definition.
 * I agree that using "Topologist's Sine Curve" is more descriptive than "Graph of Sine of Reciprocal". I'll have a look at your definition. --Anghel (talk) 16:52, 12 January 2013 (UTC)

Alternative proof
I quick glance at S & S suggests they have what looks to be a much simpler proof using local connectedness. --Dfeuer (talk) 17:02, 12 January 2013 (UTC)


 * Feel free to add it. --prime mover (talk) 22:06, 12 January 2013 (UTC)