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I'm studying two algebraic topology texts (namely Munkres and Theodore)

Here are definitions given in those texts

Munkres

Let $X$ be a topological space.

If the identity map on $X$ is null-homotopic, then $X$ is contractible.

.

Theodore

Let $X$ be a topological space and $x_0\in X$.

If there is a continuous map $F:X\times[0,1]\rightarrow X$ such that $F(x,0)=x$ and $F(x,1)=x_0$ ($x \in X$) and $F(x_0,t)=x_0$ ($t\in[0,1]$), then $X$ is contractible.

You can see that Theodore's definition is stronger than Munkres', since $F$ in the Theodore's definition is a homotopy relative to $\{x_0\}$

Which is the standard definition for a contractible space?

And is a contractible space simply connected under the Munkre's definition?

I saw posts here saying that "Contractible space is simply-connected".

However, with Munkres definition, I can only show that two paths are homotopic, not path-homotopic. How do I show they are path-homotopic?

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    $\begingroup$ It's actually a nice exercise to prove that both notions are equivalent! :) A map is homotopic to a constant map iff it's homotopy to a constant map rel a point. (Of course it only work with constant) So there isn't really a "standard definition" between the two, because they're the same. $\endgroup$ – Najib Idrissi Oct 17 '14 at 18:45
  • $\begingroup$ @NajibIdrissi That sounds amazing. Would you please show me how to prove the equivalence? $\endgroup$ – Number 9 Oct 17 '14 at 18:47
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    $\begingroup$ I don't really have time or energy right now, maybe tomorrow if nobody has answered. Now that I think of it it's possible that this result is true only in reasonable spaces (say CW complex, or at least well-pointed?). $\endgroup$ – Najib Idrissi Oct 17 '14 at 18:51
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    $\begingroup$ @NajibIdrissi I don't think they are equivalent in general. Chapter 0 in Hatcher's book has a couple of examples of spaces such that the identity map is nullhomotopic, but not nullhomotopic relative to a point. $\endgroup$ – Ayman Hourieh Oct 17 '14 at 19:07
  • $\begingroup$ @AymanHourieh Yes, that's what I thought. That space isn't well-pointed though. $\endgroup$ – Najib Idrissi Oct 18 '14 at 7:21
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These two definitions are not equivalent in general. In my experience, the first definition is more common. For an example of a space whose identity is nullhomotopic but not nullhomotopic relative to a point, see exercises 6 and 7 in chapter 0 of Hatcher's Algebraic Topology.

If a space is contractible in the sense that the identity is nullhomotopic, then it is homotopy equivalent to a point. In particular, it is simply connected.

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