In class, Invariance of Domain was proved using lots of machinery, but I came up with the following 'proof' whose truth I am suspicious of.
Explicitly statement to prove is :
Let $B$ = $\{ \vec{x} \in \mathbb{R^2} \hspace{2mm} : \|\vec{x}\| < 1 \}$ and $f:B \rightarrow f(B) \subset \mathbb{R^2}$ be an injective continuous map with vector outputs (in the plane). Then f is a homeomorphism. $B$ is given the subspace topology.
Alleged proof goes like this:
Let $B_r = \{ \vec{x} \in \mathbb{R^2} : \|\vec{x}\| \leq r \}$ be any closed ball of radius $r<1$ inside $B$. Restriction of $f$ to $B_r$ is injective map from a compact space to Hausdorff space, hence is homeomorphism. In particular, it restricts to a homeo between $O_r = \{ x\in B_r : \|\vec{x} \|<r \}$ of $B_r$ , which is an $r$-ball in $\mathbb{R^2}$, and its image. Notice $r$ was almost arbitrary (apart from having proper size).
(I couldn't connect this two sections into $1$ section. If you could explain how to, please let me know)
Now take any point $y$ in $f(B)$. We must have $y = f(x)$ for some $x\in B$ and we can choose a small enough neighborhood around $x$ with radius $\epsilon$ so that $f$ on $B_{\| x \| + 2\epsilon}$ is homeomorphism. Thus it sends $\epsilon$ ball around x into another ball around $y$ which lies in $f(B_{\| x \| + 2\epsilon}) \subset f(B)$.
Is there something wrong here? It should not have been this easy.