This is a general topology question.
Let $k < n$ be positive integers. Suppose we have opens $U \subset \mathbf R^k$ and $V \subset \mathbf R^{n-k}$ and a continuous and injective map $$f: U \times V \hookrightarrow \mathbf R^{n}.$$
I cannot think of an example of such $U,V,f$ such that the image of $f$ is not an open (in $\mathbf R^{n}$) but I see no reason why it should be open. I am very inclined to think that there are examples were $\mathrm{im}(f)$ is not open.
Could you please provide such an example?
[I first thought that a fattened version of the usual "non-submanifold" "looking like a $\sigma$" would do the job, but it does not seem to.]
There are no such examples, though it is not trivial to prove. It is a classical result by Brouwer: Invariance of domain