If $(X,\tau)$ is a paracompact, is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?

up vote 7 down vote accepted

Any discrete space is paracompact, since the family of singletons is locally finite and an open refinement of every open cover. Put $\tau'=\mathcal{P}(X)$. Then the discrete space $(X,\tau')$ is paracompact, and $\tau'$ is maximal with respect to inclusion and paracompactness.

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