Comments on: Connectedness
https://thetwomeatmeal.wordpress.com/2010/11/09/connectedness/
and the Two Meat MealThu, 30 Aug 2012 02:33:03 +0000
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By: DMG
https://thetwomeatmeal.wordpress.com/2010/11/09/connectedness/#comment-319
Thu, 30 Aug 2012 02:33:03 +0000http://thetwomeatmeal.wordpress.com/?p=157#comment-319You say that connected components are clopen, which is not true in general. It is true that they are closed, which follows from the fact that the closure of a connected subset is also connected and the maximality of connected components. But for example consider the set of rational points in R (subspace topology), the connected components are points (because Q are dense), and they are not open!
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By: Countability Axioms « Gracious Living
https://thetwomeatmeal.wordpress.com/2010/11/09/connectedness/#comment-108
Fri, 31 Dec 2010 06:55:32 +0000http://thetwomeatmeal.wordpress.com/?p=157#comment-108[…] far we’ve seen two basic families of properties of topological spaces. The connectedness axioms tell us in what ways it is possible to break our space apart into pieces. The compactness […]
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By: Subgroups of Free Groups are Free « Gracious Living
https://thetwomeatmeal.wordpress.com/2010/11/09/connectedness/#comment-104
Tue, 28 Dec 2010 12:33:18 +0000http://thetwomeatmeal.wordpress.com/?p=157#comment-104[…] is connected if there’s a path between any two of its vertices: equivalently, if it’s path-connected as a topological space. A cycle is just a path from a vertex to itself. The most important kind of graph, for us, is a […]
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By: Differential Geometry and the Sphere Theorem « Gracious Living
https://thetwomeatmeal.wordpress.com/2010/11/09/connectedness/#comment-96
Fri, 17 Dec 2010 01:39:11 +0000http://thetwomeatmeal.wordpress.com/?p=157#comment-96[…] explanation: “connected” means what you think it means. ”Simply-connected” means that every loop can be contracted to a point; on a torus, […]
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By: Local Connectedness and Some Weird Counterexamples « Gracious Living and the Two Meat Meal
https://thetwomeatmeal.wordpress.com/2010/11/09/connectedness/#comment-33
Wed, 10 Nov 2010 14:26:05 +0000http://thetwomeatmeal.wordpress.com/?p=157#comment-33[…] time we talked about connectedness and path-connectedness and showed that though all path-connected spaces are also connected, connected spaces are not in […]
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