TITLE: Path connected but not metrizable
QUESTION [5 upvotes]: 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)?

REPLY [2 votes]: $\pi$-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