If X is a compact metric space then,there exists a surjective function $\mathbb R \to X$
I think the above statement is false. But I can't think of a counter example.
Any help?
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simu tiyam
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2Are you looking for a continuous function? – jjagmath Apr 18 '21 at 13:24
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No, I am no looking for continuous function. – simu tiyam Apr 18 '21 at 13:34
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I suggest you read about the Hahn–Mazurkiewicz theorem. – Moishe Kohan Apr 21 '21 at 18:25
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Any continuous image of $\Bbb R$ is (path-)connected. So no such function exists for $\{0,1\}$ (which is finite, so compact) or the Cantor set e.g.
If the function need not be continuous, then note that any compact metric space has size at most $|\Bbb R|$, being separable. So then one can do it.
Henno Brandsma
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I am not looking for continuous function. It can be arbitrary surjection function. – simu tiyam Apr 18 '21 at 13:30
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1@simutiyam A compact metric space is separable. A separable metric space has size at most $|\Bbb R|$. – Henno Brandsma Apr 18 '21 at 13:38