About the definition of homotopic equivalence.

Question

Are the definition for homotopy relation between a pair of path and the definition for homotopy relation for the topological spaces are different? Thanks.

Answer 1

You can shrink the disc continuously along radii to the center. Homotopic equivalence doesn't mean the spaces are the same (homeomorphic). These two obviously aren't.

Here's the definition from Wolfram:

Two topological spaces $X$ and $Y$ are homotopy equivalent if there exist continuous maps $f:X \rightarrow Y$ and $g:Y \rightarrow X$, such that the composition $f \circ g$ is homotopic to the identity $\text{id}_Y$ on $Y$, and such that $g \circ f$ is homotopic to $\text{id}_X$. Each of the maps $f$ and $g$ is called a homotopy equivalence, and $g$ is said to be a homotopy inverse to $f$ (and vice versa).

This is an equivalence relation: it's clearly symmetric in $X$ and $Y$.

In your question, take $f$ to be the contraction along radii and $g$ to be the identity mapping the center to itself.

Wolfram continues

Some spaces, such as any ball $B^k$, can be deformed continuously into a point. A space with this property is said to be contractible, the precise definition being that $X$ is homotopy equivalent to a point. It is a fact that a space $X$ is contractible, if and only if the identity map $\text{id}_X$ is null-homotopic, i.e., homotopic to a constant map.