Path connected but not metrizable What are the examples of path connected spaces which are not metric spaces. The only examples I know are sets with indiscrete topology?
Are there such spaces which are not simply connected (the indiscrete space is simply connected)?
 A: One can take familiar spaces with different topologies. For example, one could take $\mathbb{C}$ with the Zariski topology (ie cofinite topology). If you recall that metric spaces are always Hausdorff, then this yields an example because $\mathbb{C}$ with the Zariski topology is not Hausdorff (good exercise).
A: A classic example is the long line (and its subspaces the open and closed long rays). It's locally homeomorphic to $(0,1)$, path-connected, and hereditarily normal, so it's a pretty nice space, but it's sequentially compact without being compact, so it's not metrizable.
A: Take any path connected space and take its Cartesian product with uncountably many copies of $\mathbb{R}$. This space is path connected but not metrizable and is homotopy equivalent to the original space, so in particular it can have nontrivial fundamental group.
A: Take countably many disjoint copies of the unit interval, say $I_n=[0,1]\times\{n\}$. 
Glue together all points $\langle 0,n \rangle$ into one point, $p$. (Formally take a quotient map, where $\langle 0,n \rangle\sim \langle 0,m \rangle$ for all $n,m$, and take the quotient topology). The result (also known as the non-metric hedgehog) is not first-countable at $p$. But every two points could be connected with a path, which would necessarily go through $p$ if the two points come from different copies of the unit interval.   
A: $\pi$-Base is an online database of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following twenty-seven path-connected, non-metrizable spaces. You can learn more about any of these spaces by viewing the search result.
Alexandroff Square
Compact Complement Topology
Countable Excluded Point Topology
Countable Particular Point Topology
Deleted Diameter Topology
Deleted Radius Topology
Divisor Topology
Double Origin Topology
Finite Complement Topology on an Uncountable Space
Finite Excluded Point Topology
Finite Particular Point Topology
Half-Disc Topology
Indiscrete Topology
Nested Interval Topology
Niemytzki's Tangent Disc Topology
One Point Compactification Topology
Overlapping Interval Topology
Prime Ideal Topology
Radial Interval Topology
Right Order Topology on $\mathbb{R}$
Sierpinski Space
Simplified Arens Square
Telophase Topology
The Integer Broom
The Long Line
Uncountable Excluded Point Topology
Uncountable Particular Point Topology
