A closed connected component in a topological space does not contain any path-connected subset? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:11:44Z http://mathoverflow.net/feeds/question/109116 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109116/a-closed-connected-component-in-a-topological-space-does-not-contain-any-path-con A closed connected component in a topological space does not contain any path-connected subset? Changyu Guo 2012-10-08T06:25:08Z 2012-10-08T12:30:57Z <p>Does there exist such a non-trivial closed connected component U of some connected topological space X or a non-trivial connected topological space X that do not contain any non-trivial path-connected subset.</p> <p>The answer is negative if the space is assumed to be connected and locally path-connected. Since every component of a connected and locally path-connected space is path connected.</p> <p>Added after some useful comments: If we assume that the space X is actually a metric space (together with the metric topology), then can it possible to contain non-trivial path-connected subset. Note that i assume that any component of the metric space X is non-trivial (not a point).</p> http://mathoverflow.net/questions/109116/a-closed-connected-component-in-a-topological-space-does-not-contain-any-path-con/109140#109140 Answer by Mark Grant for A closed connected component in a topological space does not contain any path-connected subset? Mark Grant 2012-10-08T12:30:57Z 2012-10-08T12:30:57Z <p>What you are asking for is a connected and totally path-disconnected space. Apparently there is such a beast on page 145 of "<a href="http://www.amazon.co.uk/Counterexamples-Topology-Dover-Books-Mathematics/dp/048668735X#reader_048668735X" rel="nofollow">Counter-examples in Topology</a>" by Steen and Seebach (I don't have a copy of the book, and the page in question is missing from the linked preview). It is amusingly called "Cantor's Leaky Tent" and is even a subspace of \$\mathbb{R}^2\$.</p> <p>See also <a href="http://mathoverflow.net/questions/23430/what-is-an-example-of-a-non-regular-totally-path-disconnected-hausdorff-space" rel="nofollow">http://mathoverflow.net/questions/23430/what-is-an-example-of-a-non-regular-totally-path-disconnected-hausdorff-space</a></p>