Manifold and Weak Topologies Coincide

Question

Let $M$ be a manifold and $X_i$ a topological space with $f_i: M \rightarrow X_i$ for each $i \in I$. The weak topology on $M$ induced by $\{f_i\}$ is the coarsest topology on $M$ for which $f_i$ is continuous for each $i \in I$. The collection of sets of the form $f_i^{-1}[U_i]$ for $U_i$ open in $X_i$ for all $i \in I$ is a subbase for the weak topology on $M$. If the manifold topology on $M$ coincides with the weak topology on $M$, is this subbase also a base?