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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.

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.

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).

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@Changyu Guo: I think your question needs to be modified slightly, since a point is path-connected. –  Mark Grant Oct 8 '12 at 6:28
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Wouldn't any countable connected space have this property (with U=X)? –  Goldstern Oct 8 '12 at 7:34
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@Changyu Guo: If by "non-trivial" you mean "containing more than one point", then the one-point space is still problematic: it is a connected space which does not contain any "non-trivial" subsets! –  Mark Grant Oct 8 '12 at 8:51
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@Changyu Guo: See for instance ams.org/journals/proc/1970-026-02/S0002-9939-1970-0263005-0/… and the references therein for the spaces Goldstern had in mind. –  Tapio Rajala Oct 8 '12 at 9:55
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Consult Steen & Seebach for all types of topological counterexamples. The automated index is "Spacebook" at austinmohr.com/home/?page_id=146 ... set it for "connected but not path connected" to get the list of examples, and then if necessary examine these examples in the Steen & Seebach book itself. –  Gerald Edgar Oct 8 '12 at 12:29
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1 Answer

up vote 5 down vote accepted

What you are asking for is a connected and totally path-disconnected space. Apparently there is such a beast on page 145 of "Counter-examples in Topology" 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 $\mathbb{R}^2$.

See also What is an example of a non-regular, totally path-disconnected Hausdorff space?

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