I want to know the following is well-known or not:
Let X be a metric space with Hausdorff dimension \alpha. $\alpha$. Then for any \beta $\beta < \alpha, alpha$, X contains a closed subset whose Hausdorff dimension is \beta.$\beta$.
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Question on geometric measure theoryI want to know the following is well-known or not: Let X be a metric space with Hausdorff dimension \alpha. Then for any \beta < \alpha, X contains a closed subset whose Hausdorff dimension is \beta.
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