Let us denote $R$ to be a rectangle, i.e. topological boundary of a set of the form $[a,b]\times[c,d]$ in $\mathbb R^2.$ I want to show that given any rectangle $R,$ there does not exist any topology on $R$ and smooth structure on it such that the natural inclusion map $i:R\to\mathbb R^2$ is not an injective immersion.
I can definitely solve this if the topology of $R$ is the subspace topology induced by $\mathbb R^2$. Then I can choose a corner of the rectangle and show that the tangent space is two dimensional at that point, which is not possible. How to solve the general case?