Question
Is $f(x)=(2x-1)/3:\Bbb Z_2\to\Bbb Z_2^\times$ a near-homeomorphism? Semihomeomorphism? Homeomorphism?
My Attempt
I have that $f(x)=(2x-1)/3:\Bbb Z_2\to\Bbb Z_2^\times$ is continuous and has a continous inverse because $\lvert f(x)-f(y)\rvert_2=\frac12\lvert x-y\rvert_2$
I have that $f$ can't be a homeomorphism $\Bbb Z_2\to\Bbb Z_2$ because it doesn't surject.
What about $\Bbb Z_2\to\Bbb Z_2^\times$ though? Which is the question.
Looks to me like it's sufficient to prove $f,f^{-1}$ both inject. Both are functions so have an injective inverse, it just remains to prove they land inside the domain and range.
$f^{-1}(\Bbb Z_2^\times)\subseteq\Bbb Z$ because $\lvert f^{-1}(x)\rvert_2\leq0$
and
$f(\Bbb Z_2)\subseteq\Bbb Z_2^\times$ because $\lvert(2x-1)/3\rvert_2=1$
So it's a homeomorphism from $\Bbb Z_2\to\Bbb Z_2^\times$, correct?