2
$\begingroup$

If a space is hyperconnected (that is, the every non-empty open sets intersect), is it also path-connected?

$\endgroup$
7
$\begingroup$

No - take $(\mathcal{P}(\mathbb{N}), \tau)$ where $\tau = \{\emptyset\}\cup\{A\subseteq\mathbb{N}: \mathbb{N}\setminus A \text{ is finite}\}$. Clearly, every two non-empty open sets have non-empty intersection, so the space is hyperconnected, but is not path-connected, see this post.

$\endgroup$

Your Answer

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