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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)?

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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.

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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).

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  • $\begingroup$ why is this space path connected? I know it is connected. $\endgroup$ – Arun Kumar Dec 4 '14 at 6:02
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    $\begingroup$ Dear @ArunKumar: Take any two points, the usual interval between them in $\mathbb{C}$ with the usual topology yields a map $[0,1]\rightarrow \mathbb{C}$. Now just observe that this map is continuous in the Zariski topology because the inverse image of closed set (ie a finite set of points) is either a finite set of points in $[0,1]$ or the empty set, both of which are closed. Hence it is path connected. $\endgroup$ – Moss Dec 4 '14 at 6:23
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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.

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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.

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$\pi$-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

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