I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to $\mathbb{R}^{2m+1}$).
I don't know how dumb this question is, but in what sense is it open? In which topology do we have this openness of our set of interest? I'm trying to develop an intuition on the set of such mappings, so I ask this question.
It's a good question. Usually for mapping spaces, one uses the compact-open topology. It has as a subbase consisting of sets $C(K, U) = \left\{f: X \to Y \;|\; f(K) \subseteq U \right\}$, where $K$ varies over compact $K \subseteq X$ and open $U \subseteq Y$. This topology enjoys many natural properties (especially when the spaces are Hausdorff) that are itemized on the wiki page.
Exactly what constitutes a map $f:X \to Y$ depends on the category that you're interested in. It's likely that you're considering smooth maps, given the (differential-topology) tag.