Clarification on understanding of retracts and deformation retracts I've been trying to get a better intuitive understanding of retracts and deformation retracts. I recently found this answer on Quora, which helped a lot.
I like the intuition in the above link involving thinking of the space $X$ as a city, with a deformation retract $A$ being a park in the city, or something of the like, whilst elements of $X$ are "people". I'm going to ask my question in that context, but if anybody has any better intuition that is also welcome.
For $A$ to be a retract, people in $X$ are allowed to "teleport" to $A$. As long as people who are "close" in $X$ end up "close" in $A$, and people who are already in $A$ stay where they are, we've got a retract. This makes a lot of sense.
For $A$ to be a deformation retract on the other hand, people in $X$ actually have to "walk" to their destination in $A$, in a continuous way (and all the other rules about people in $A$ ending up where they start off must also be obeyed).
If I understand correctly, given a path connected space $X$ (say $T^2$), any point in $X$ is a deformation retract of $X$. In the case of $T^2$, In order for everybody to walk to a point, at least some people who are near to each other at the start must at some point take different paths around the hole in the middle of the $T^2$, so even if they end up near each other (at a point $x$), they have during the process gone away from each other, but this must be OK.
Is my understanding above is correct? Somehow the people "going away from each other" during their "walking" is upsetting me, but I can't really explain why.
 A: You’re good up to the point where you say that any point is a deformation retract in a path-connected space. This isn’t necessarily true. Take the circle $S^1$ for example. It can be shown that the identity is not homotopic to a constant map (which is the same thing as saying that there isn’t a deformation retraction onto a point). The “points walking away” example is exactly stating that the retraction isn’t continuous if you tried. I see it as having to tear $S^1$ To make the deformation work, but that makes it discontinuous.
This picture of points walking away is useful but unfortunately doesn’t work well in a proof. But you can see it nicely in other cases too! Take the disc $D^2 \subset \mathbb{R}^2$. You can see that any retraction from the disc onto it’s boundary circle wouldn’t be continuous because it would ‘tear’ a hole somewhere. There are a lot of theorems about this. In fact, a manifold can’t retract onto its boundary, and using this you can prove neat results like Brouwer’s fixed point theorem.
