Finite partition of a manifold into manifolds

Question

Let $X$ be a topological $n$-manifold and $C \subset \mathcal{P}(X)$ a finite partition of $X$ into topological manifolds (with subspace topologies). Does there exist at least one $M \in C$ such that $\dim(M) = n$?

Answer 1

Yes (assuming $X$ is nonempty), and this holds even for countable partitions, or even for countable covers whose elements may not be disjoint. Covering each element of $C$ by countably many compact subsets, the Baire category theorem says that some $M\in C$ must have nonempty interior in $X$. Thus an $n$-ball embeds in $M$ and so $\dim M$ must be $n$.

(Here I assume your manifolds are required to be second-countable, but if not, you can just restrict everything to a single coordinate patch on $X$.)