Connected But Not Path-Connected? Can you think of any spaces that are connected but not path connected apart from the Topologist's Sine Curve? 
 A: An example of a connected space that is not path-connected is the deleted comb space:
$$ (\{0\} \times \{0,1\}) \cup (K \times [0,1]) \cup ([0,1] \times \{0\})$$
where $K = \{ \frac{1}{n} \mid n \in \mathbb{N} \}$
Taken from here.
A: Another standard example is the extended long line. Counterexamples in Topology will have more, but my copy isn't to hand right now.
A: The canonical example is the extended long line. You can think of the regular line $[0,\infty)$ as the product of $[0,1)$ and $\omega$ in the dictionary order topology—effectively, a countable number of copies of $[0, \infty)$ pasted end-to-end. 
The long line is the same way, except that instead of a countable number of copies you use an uncountable number of copies: take $[0, 1)\times\omega_1$ in the dictionary order topology, where $\omega_1$ is the smallest uncountable ordinal. Then to get the extended long line, you add one more point $p$ onto the far end.  It's clearly connected, but it isn't path-connected because the path from any finite point, say $(1/2, 1)$, is too far from $p$ for the path between them to be the image of $[0,1]$.
The book Counterexamples in Topology by Seebach and Steen is good for answering questions like this.
A: Here are a whole bunch from $\pi$-Base, a searchable version of Steen and Seebach's Counterexamples in Topology. You can visit the search result to learn more about any of these spaces.
An Altered Long Line
A Pseudo-Arc
Cantor's Leaky Tent
Closed Topologist's Sine Curve
Countable Complement Extension Topology
Countable Complement Topology
Double Pointed Countable Complement Topology
Finite Complement Topology on a Countable Space
Gustin's Sequence Space
Indiscrete Irrational Extension of $\mathbb{R}$
Indiscrete Rational Extension of $\mathbb{R}$
Irrational Slope Topology
Lexicographic Ordering on the Unit Square
Nested Angles
One Point Compactification of the Rationals
Pointed Irrational Extension of $\mathbb{R}$
Pointed Rational Extension of $\mathbb{R}$
Relatively Prime Integer Topology
Roy's Lattice Space
Smirnov's Deleted Sequence Topology
The Extended Long Line
The Infinite Broom
The Infinite Cage
Topologist's Sine Curve
A: My favourite example is the Solenoid. It is also a topological group, which usually don't admit such weird pathological properties. 
