Map between manifolds. Let $M,N \subset \mathbb{R}^3$ be (not necessarily smooth) 2-manifolds without boundary. Let $f: M \rightarrow N$ be a continuous function and suppose that $f$ is injective. Let $x \in M$ and let $U$ be an open neighborhood containing $x$. Suppose that $f(U) \subset N$ has a boundary and $f(x)$ is on the boundary of $f(U)$. Does this violate the assumption that $f$ was injective?

Take a compact neighborhood $K$ of $x$ ($U\subset K$).
Since $Y$ is Hausdorff, the map $f|_{K}:K\to f(K)\subset Y$ is a homepmorphism.
Then $f|_{U}$ is also a homeo.
On the otherhand, we can prove that there is no homeomorphism between the 2-disk and some area in $\mathbb R^2$ with boundary
by the homology theory.
