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.
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Martin Sleziak
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user156619
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2What about $[0,1]^\Bbb{R}$? – PhoemueX Sep 10 '14 at 17:58
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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.
user156619
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A compact example (containing this as a subspace!) is the Double Arrow space, see http://math.stackexchange.com/a/75495/4280. – Henno Brandsma Sep 10 '14 at 21:12
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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).
Moishe Kohan
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