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  1. Prove that any metrizable compactum is a continuous image of a Cantor perfect set.
  2. What is the cardinality of a compact metrizable without isolated points?
  3. Is the space “two arrows of Alexandrov” a continuous image of a Cantor perfect set?

"two arrows of Alexandrov" = $\{0\}\times(0,1]\cup\{1\}\times[0,1)$.

I know that the first question can be found in some books but I can't find the proof

Cl0ser
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    Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Dec 29 '20 at 08:38
  • 2 is continuum (size of $\Bbb R$) of course. See the Cantor set and $[0,1]$. 3 is no, as the continuous image of a metric compact I’m is still metrisable. 1 has many proofs and will depend on your background and known theory you should clarify what theorems you already covered before this problem. – Henno Brandsma Dec 29 '20 at 13:31
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    The answers here outline a proof of (1). This answer answers (2). (1) answers (3) once you prove that the double arrow space is not metrizable; you can do this by showing that it is not second countable. – Brian M. Scott Dec 29 '20 at 19:46

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