Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic

Question

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic.($|X|>1)$

Please suggest me ways on how should I think about this.Its quite sure that $X$ cant be finite.

I tried discrete topology where the conditions got well except its compact.

Answer 1

Any one-point space (which is finite, yes) clearly verifies this... And now, you're going to edit your question and say : "please help to find a compact $X$ that is not one point, such that $X$ and $X\times X$ are homeomorphic"...

Answer 2

$X=\{0\}$ works, doesn't it? Both $X$ and $X\times X$ have only one element.

Answer 3

Hint: look at infinite powers $X^I$ of finite discrete spaces $X$, e.g. $X=\{0,1\}$. This provides non-trivial Hausdorff examples. To get a metric example you need to take a little bit of care on (the cardinality of) the index set.

Edit: $X$ need not be finite but can be any compact metric space.