Lemma: Let $E$ be a closed bounded subset of $\Omega$ and let $T$ be an injective transformation of class C' on $\mathbb{R}^3$. Also the Jacobian does not vanish. Define $v(K)$ to be the volume of $K.$ Then $\lim_{C\downarrow p} \frac{v(T(C))}{v(C)} = J(p)$ where $C$ ranges over the family of cubes lying in $\Omega$ and having center $p$, and the limit is uniform for all $p\in E.$
In the proof of the above lemma, it says
If $p$ lies in the sphere $S$ with center $p_0$ and radius $r<\epsilon$, then $T(p)$ lies in the sphere whose center is $T(p_0)$ and whose radius is $(1+\epsilon)r.$ Moreover, when $p$ lies on the boundary of $S$, $T(p)$ lies outside the sphere with center $T(p_0)$ and with radius $(1-\epsilon)r.$
That stuff is fine and I understand the derivations which are in the book.
However, it then says
Since $T$ is injective and takes open sets into open sets, we see that $T(S)$ must contain the smaller sphere of radius $(1-\epsilon)r$.
I do not understand why injectivity and taking open sets to open sets implies that $T(S)$ contains the smaller sphere. That is where I am stuck.
Thanks.
And why did this get downvoted??