A space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure?

Asked May 11 '22 at 23:03

Active May 11 '22 at 23:03

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Question :

Does there exist a space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure ?

My attempt :

The problem is I have no idea whether it is true or false. If I were to prove it true, I think I would look at Cantor spaces or similar. If it was false, I would probably some uniform continuity on compact neighbourhoods.

Can someone help me ?

asked May 11 '22 at 23:03

math

2,313

1

The $x$-axis has measure $0$ in the plane, but it is homeomorphic to the real line, which has measure $\infty$ in itself. But maybe you wanted both spaces to be subspaces of $\mathbb{R}^1$? – Alex Kruckman May 11 '22 at 23:36
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Do you know that all Cantor sets are homeomorphic to each other? – Dave L. Renfro May 11 '22 at 23:44
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A "fat Cantor set" is homeomorphic to the Cantor set but has positive measure. – DanielWainfleet May 12 '22 at 06:50
@DaveL.Renfro Why is true ? Do you have a proof of it ? – math May 12 '22 at 13:03
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See Homeomorphism between two Cantor sets AND Homeomorphism between cantor sets one with measure 0 and one with positive measure AND The Cantor set is homeomorphic to infinite product of ${0,1}$ with itself - cylinder basis - and it topology. – Dave L. Renfro May 12 '22 at 13:31

A space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure?

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