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).

  • 7
    $\begingroup$ @Changyu Guo: I think your question needs to be modified slightly, since a point is path-connected. $\endgroup$
    – Mark Grant
    Oct 8 '12 at 6:28
  • 5
    $\begingroup$ Wouldn't any countable connected space have this property (with U=X)? $\endgroup$
    – Goldstern
    Oct 8 '12 at 7:34
  • 1
    $\begingroup$ @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! $\endgroup$
    – Mark Grant
    Oct 8 '12 at 8:51
  • 2
    $\begingroup$ @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. $\endgroup$ Oct 8 '12 at 9:55
  • 2
    $\begingroup$ 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. $\endgroup$ Oct 8 '12 at 12:29

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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.