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).



A closed connected component in a topological space does not contain any path-connected subset?
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](http://www.amazon.co.uk/Counterexamples-Topology-Dover-Books-Mathematics/dp/048668735X#reader_048668735X)" 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?](https://mathoverflow.net/questions/23430/what-is-an-example-of-a-non-regular-totally-path-disconnected-hausdorff-space)


