For compact sources the immersion condition is open in the weak smooth topology

Empty $M$ and $m=0$ have automatic injectivity; if $M$ is nonempty and $m>n$, both loci are empty and open. For (i) in the remaining case, fix an arbitrary immersion $f:M\to N$. Around each source point take a source chart mapped by $f$ into a target chart, and a smaller compact coordinate ball whose interior contains that point. Compactness selects finitely many such pieces $K_i\subset U_i$ covering $M$, with $f(U_i)\subset V_i$. Write $J_i(x)=D({\psi }_i\circ f\circ {\phi }_i^{-1})({\phi }_i(x))$; its entries are continuous and bounded on $K_i$.

For each $i$ and $x\in K_i$ some $m\times m$ minor of $J_i(x)$ is nonzero by [L1], so the maximum ${\delta }_i(x)$ of the absolute determinants of the finitely many minors is a continuous strictly positive function on $K_i$; by [L3] it has a positive minimum $c_i$. Let $B_i\ge 1$ bound the absolute values of all entries of $J_i$ on $K_i$. The determinant of an $m\times m$ matrix is a polynomial in the entries, so there is ${\eta }_i>0$, depending only on $m$, $B_i$, $c_i$, such that any matrix $J^{\mathrm{\prime }}$ with $\mathrm{\mid }{J}^{\mathrm{\prime }}-J_i(x)\mathrm{\mid }<{\eta }_i$ entrywise satisfies $\mathrm{\mid }\det J^{\mathrm{\prime }}-\det J_i(x)\mathrm{\mid }<c_i\mathrm{/}2$ for the maximizing minor, hence has a nonzero $m\times m$ minor. Choosing the finitely many ${\eta }_i$ uses no choice, and may be taken in the form $2^{-k}$ with the least suitable $k$.

Let $\mathcal{U}$ be the basic weak open set of maps $g:M\to N$ determined by the data $(U_i,{\phi }_i)$, $(V_i,{\psi }_i)$, $K_i$, $r_i=1$, ${\epsilon }_i={\eta }_i$. Its definition constrains the partial derivatives of first order of ${\psi }_i\circ g\circ {\phi }_i^{-1}$ on ${\phi }_i(K_i)$ to differ from those of $f$ by less than ${\eta }_i$; in particular every entry of the Jacobian of $g$ differs from the corresponding entry of $J_i$ by less than ${\eta }_i$ at every point of $K_i$. By step 2.1 every such $g$ has rank $m$ at every point of $M$, so by [L1] every $g\in \mathcal{U}$ is an immersion. Hence $\mathrm{Imm}(M,N)$ contains the basic neighbourhood $\mathcal{U}$ of $f$ and is open.

For (ii), independently fix an arbitrary fibrewise injective pair $(f,F)$, whose base map need not be an immersion. Choose compact pieces $K_i$ and induced bundle charts over source and target chart domains. In such charts a bundle map has the form $(x,v)\mapsto (g(x),A_i(x)v)$. The compact set of vectors $(x,e_j)$ with $x\in K_i$, $1\le j\le m$, is a valid compact test set in $TM$; zeroth-order control of the images of these vectors controls every column of $A_i$. Zeroth-order control of $g$ keeps these images in the same target bundle chart. Each matrix $A_i$ has rank $m$ by the injectivity of $F$, so its own maximum of absolute $m$-minors has a positive minimum on $K_i$. Apply the polynomial determinant estimate of step 2.1 to these matrices, independently of $df$. This gives a neighbourhood of the pair $(f,F)$, among all bundle-map pairs, on which all fibre maps remain injective. Restricting that neighbourhood to the fixed-base fibre proves its openness as well.