Comparing 2 solutions of problem 2 chapter 0 of Allen Hatcher.

Question

The question is to construct an explicit deformation retraction of $\mathbb{R^n} - \{0\}$ onto $S^{n-1}.$

Here is the answers I found online so far:

The first solution:

The second solution

Here are my questions:

1- Is the first answer correct? Is it proving that there is a weak deformation retract? Is the second answer correct? Is it proving that there is a strong deformation retract instead?

2- I do not understand really the difference between the definition of a weak deformation retract and a strong deformation retract when I am trying to prove them. could anyone explain this for me please?

Here is the definition of SDR from AT:

And here is the definition of WDR from Rotman:

And here is the definition of SDR from Rotman:

Could anyone help me clarify these descripencies please?

Answer 1

I think Hatcher's definition of DF is a bit stronger than the usual definition. Let's take Wikipedia as a reference -- it coincides with Rotman definition. In this sense, $f_t$ is a deformation retract if $f_t$ has properties (1), (2), (3) and property (4) for $t=1$. The SDF according to wiki-definition ("DF" for Hatcher) requires (4) for all $t$.

So yes, your first solution shows DF in the usual sense, and your second solution shows SDF if you assume "for all $t$" in point (4).