Difference between retract and deformation retract 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?

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!