A perfect Hausdorff space that is not metrizable.

Question

Can anyone provide an example of a well-known topological space that has the following three properties:
(1) It is perfect (contains no isolated points),
(2) T2, and
(3) not metrizable.

Answer 1

The so-called "Sorgenfrey line" or "lower limit topology" is an example of a Hausdorff, perfect, non-metrizable space. The topology of the Sorgenfrey line is generated by the basis of all half-open intervals $[a,b)$, where $a$ and $b$ are real numbers.

It is well known that the space is both Hausdorff and not metrizable. And here

https://dantopology.wordpress.com/tag/the-sorgenfrey-line/

there is a nice proof that the space is perfect.

Answer 2

Even more interesting example is a countably infinite connected Hausdorff space (irrational slope topology). A countably infinite connected space has to be perfect and cannot be metrizable (it has no nonconstant continuous functions).