This is part of Problem 16 in Chapter 6 of Lee's Smooth Manifolds.
Let $N,M,S$ be smooth manifolds. A smooth family of maps is a collection $\{F_s:N\to M \;|\; s\in S\}$ such that $F_s(x)=F(x,s)$ for some smooth $F:N\times S\to M.$ A class $\mathcal F$ of maps is called stable if whenever $\{F_s:N\to M \;|\; s\in S\}$ is a smooth family of maps and $F_{s_0}\in\mathcal F$ for some $s_0\in S$, then there is some neighborhood $U$ of $s_0$ in $S$ so that $F_s\in \mathcal F$ for each $s\in U.$
The problem I am asked to do is show that, under the assumption that $N$ is compact, the following classes of maps are stable: immersions, submersions, embeddings, diffeomorphisms, local diffeomorphisms, and maps which are transverse to a given properly embedded submanifold $X\subseteq M.$
I have solved the problem for immersions and submersions using the fact that maps of this form are maps which have maximal rank, and rank is lower semicontinuous. I am stuck when it comes to the other classes of maps, however. For example, when $F_{s_0}$ is a diffeomorphism, it is clear to me that under small perturbations of $s,$ the matrix for $d(F_s)_x$ will be invertible for each $x,$ since $GL_n(\mathbb R)$ is open in $M_{n\times n}(\mathbb R).$ What is not clear to me is why injectivity and surjectivity should be preserved under small perturbations. I'm having similar troubles with the other parts.
Something that I'm sure of is that compactness of $N$ is a necessary assumption for all of these classes of maps to be stable. I used this crucially in my solutions for the immersions and submersions, and I have done the next exercise demonstrating that none of these classes are stable when $N$ is not compact.
Any help or hints are appreciated.
EDIT: I have reduced the parts for the class of embeddings and diffeomorphisms to showing that injectivity is preserved under perturbations. I am unsure of how to show this.
