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?

The current answer already explains why the proposed homotopy cannot work. Let me give a geometric interpretation of the two-step homotopy on the linked answer.

2009年11月19日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

2024年1月12日 · I have just started learning topology and I'm trying to prove that $\mathbb{R}$ is contractible. Since I have just started learning topology and homotopy these concepts are really abstract to me. I have seen this and this, however, I need it more detailed and here is my try:

2019年2月24日 · The term "contractible in" doesn't exist and it was made up by your professor. We can only guess what he means.

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$\begingroup$ The statement its identity map is null-homotopic means that there exists a homotopy from the identity map to a constant map, which is exactly what this answer uses (and is exactly what is used in the answer of the duplicate post linked above). $\endgroup$

2012年12月27日 · Every contractible space X is simply connected because X is homotopy equivalent to a point. Is there a direct proof of this fact? There obviously is a (free) homotopy between any loop and the trivial loop at the base point. But how to construct a based homotopy, which is required for a loop to be trivial in the fundamental group?

2019年9月4日 · Said another way, it is an easy exercise to show that contractible types are equivalent to $\textbf{1}$, the singleton type (also on page 124 of the HoTT book), so it is the right definition. Share Cite

The previous discussion Proof that retract of contractible space is contractible. used the definition that The identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. But as to what I found, a space being contractible is simply defined as