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If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a clopen partition cover if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally finite.

Is there a paracompact space $(X,\tau)$ such that

  • $(X,\tau)$ is zero-dimensional, that is for $x,y\in X$ there is $U\subseteq X$ clopen such that $x\in U, y\notin U$, and
  • there is an open cover $\mathcal{U}$ of $X$ such that $\mathcal{U}$ does not have a refinement that is a clopen partition cover

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I claim that there are such counterexamples. Any Hausdorff space where every open cover can be refined by a partition into clopen sets is called ultraparacompact. In the paper Not every O-dimensional realcompact space is N-compact by Peter Nyikos, he shows that a certain zero-dimensional metrizable space is paracompact but not ultraparacompact and not even realcompact. That being said, there are several characterizations of which spaces are ultraparacompact. In particular, the ultraparacompact spaces are precisely the strongly zero-dimensional paracompact spaces (recall that a space is strongly zero-dimensional if its Stone-Cech compactification is zero-dimensional). I gave more characterizations of ultraparacompactness in this long answer.

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  • $\begingroup$ You meant "if its Stone-Cech compactification is zero-dimensional" $\endgroup$ – Ramiro de la Vega Jan 20 '15 at 14:05
  • $\begingroup$ Ramiro de la Vega. Thanks for pointing that out. I corrected that typo. $\endgroup$ – Joseph Van Name Jan 20 '15 at 15:44
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The metrizable space alluded to above is due to Prabir Roy, see Nonequality of Dimensions for Metric Spaces. An easier example is given by John Kulesza in An example in the dimension theory of metrizable spaces.

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