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The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question.

Let $X$ be a topological space, and let $\tilde{X}\to X$ be a CW-approximation. Given that $X$ has covering dimension $n$, can anything be said about the covering dimension of the CW-approximation $\tilde{X}$? If it's not generally true that $\dim\tilde{X}\leq\dim X$, is it possible to give explicit examples?

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Barratt and Milnor (An Example of Anomalous Singular Homology) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On the other hand, it is a closed subspace of $\mathbb{R}^{n+1}$ so its covering dimension is at most $n+1$.

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