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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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Is there any result concerning on the metric dimension of inverse limit?

To be specific, my question is as follows: Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold $\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some compat …