From *Counterexamples in Topology* by Steen and Seebach (2nd edition) example 129 page 145 we have an example of **connected and totally path-disconnected space**.
It is defined as follow:

Fix $p= (1/2,1/2)$. Let $C$ be the Cantor set in the unit interval $[0,1]$. Let $E \subset C$ be the subset of $C$ that is the endpoints of the removed intervals. Let $F = C \setminus E$.
Let $L( c)$ be the line segment connecting the point $c \in C $ to $p$.
Define $Y_E=\{(x, y)\in L( c)~|~ c \in E, y \in \mathbb Q \}$ and $Y_F=\{(x, y)\in L( c)~|~ c \in F, y \not\in \mathbb Q \}$
We define **Cantor's Leaky Tent** or **Cantor's Teepee** as $Y =Y_E \cup Y_F\subset \mathbb{R}^2$ with the subspace topology.

I have two questions:

- Is there another "well-know" example of connected and totally path-disconnected space?
- Cantor's Teepee is not compact. Can we construct a non trivial compact, connected and totally path-disconnected space?

toupee? -click- Oh... well, that's disappointing :) $\endgroup$ – Bruno Stonek May 25 '14 at 17:43