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I have a trouble with distinguishing retraction and deformation retraction intuitively.

That is, deformation retraction is informally an operation on a space which continuously deform(for an example, expansion of a hole in a ball or compression toward $A$ so that $A$ is fixed) a space while a subspace $A$ is not affected by this action.

This does help a lot to visualize deformation retractions.

However, I think this kind of visualization does not distinguish retract and retraction. Retraction is a continuous function $f:X\rightarrow A$ which fixes $A$. This can be also thought of an action which continuously deform $X$ to $A$.

How do I distinguish these two? How strong deformation retraction is than just retraction?

Rubertos
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    A torus $S^1 \times S^1$ retracts onto either of those component circles. It does not deformation retract onto either. This is a typical example: all you have is a map onto a subspace, not a map and a deformation of the identity to your map. –  Jul 19 '15 at 00:58
  • @MikeMiller Oh now I get it :) would you write that as an answer? – Rubertos Jul 19 '15 at 01:01

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A torus $S^1 \times S^1$ retracts onto either of those component circles $S^1 \times \{pt\}$ or $\{pt\} \times S^1$. It does not deformation retract onto any of them. Even more extreme is that every space retracts onto any point contained in it, but won't deformation retract onto it - most spaces aren't contractible.

These are typical examples: all you have is a map onto a subspace, not a map and a deformation of the identity map to your map.

Perhaps the most important point here is: a deformation retract is automatically a homotopy equivalence. As in the example above, a retract rarely is!