If $(X,\tau)$ is a [paracompact](https://en.wikipedia.org/wiki/Paracompact_space), is there a topology $\tau'\supseteq \tau$ such that $(X,\tau')$ is still paracompact, and $\tau'$ is maximal with respect to $\subseteq$ and paracompactness?



Is every paracompact topology contained in a maximal paracompact topology?
Any [discrete](https://en.wikipedia.org/wiki/Discrete_space) space [is paracompact](https://proofwiki.org/wiki/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. 


