Question about homotopy equivalence

Question

I have this proof but I don't understand why $i\circ j$ induces a homotopy equivalence, and how to see $j_*$ is injective at the level of homology?

$X$ is a Banach space

Answer 1

The fact that $i\circ j: B_\infty X^-\hookrightarrow X\backslash X^+$ is a homotopy equivalence follows from the fact that both $$X^-\backslash \{0\}\hookrightarrow X\backslash X^+$$ and $$ B_\infty X^-\hookrightarrow X^-\backslash \{0\}$$ are homotopy equivalences. In fact the smaller spaces can be seen as deformation retracts of the larger ones.

Edit: Press $X\backslash X^{+}$ in the direction of $X^-$ to get the deformation retract of $X\backslash X^+$ to $X^-\backslash\{0\}$. That is $$x^++x^-\mapsto x^++tx^-.$$ Similarly for the other deformation retract, any $x\in X^-$ such that $|x|<R$ can be radially pushed towards the sphere radius $R$ in $X^-$ (which is the boundary of $B_\infty X^-$).