Basically, the question title is what I'm curious about. Suppose you have a sequence of orientation-preserving homeomorphisms $(f_n)_n$ from an orientable connected manifold $X$ to another manifold $Y$ of the same finite dimension; if necessary and/or convenient, one may additionally assume that $X$ is compact and with (non-empty) boundary. Furthermore, assume that $X$ and $Y$ carry metric structure, i.e. there are metrics $d_X$ and $d_Y$ on $X$ and $Y$, respectively, so that we're able to talk about uniform convergence (without any further need of mathematical tools). Now suppose that the sequence $(f_n)_n$ converges uniformly on $X$ to a homeomorphism $f: X \longrightarrow Y$. Is it true that the limit mapping $f$ is orientation-preserving as well?
In other words: I'd like to know if (under the appropriate circumstances) the property "orientation-preserving" has the same persistence property under uniform convergence as continuity, for it is well-known that the uniform limit of continuous mappings is again continuous.
I really searched for quite a bit of time now in order to find a (partial) answer to this question, but unfortunately was not able to find any reference. Actually, the question is motivated from a more concrete situation stated here. Any help, reference or hint is highly appreciated!