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:


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



Beyond Cantor's Teepee
I happened to run across an answer to 2. (and therefore 1.) just by clicking on one of the related MO links, finding the same question in a comment by Timothy Gowers under [MO16578](https://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected) that was later answered in a comment by Anton Petrunin, and then googling for more information. The key term you want seems to be "[pseudo-arc](http://en.wikipedia.org/wiki/Pseudo-arc)", apparently a construction very familiar to the point-set topology community. 


A pseudo-arc is a continuum $X$ (a connected compact metrizable space) with more than one point such that no subcontinuum $A$ (a subspace that is a continuum) is a union of two proper subcontinua of $A$. Remarkably, all pseudo-arcs are homeomorphic to one another! 


