Determine whether the following spaces \(X\) admit a retraction and/or a deformation retraction onto the designated subspace \(Y\):
\(X=[0,1]\), \(Y=\{0\}\).
\(X=[0,1]\), \(Y=\{0,1\}\).
\(X=S^1\), \(Y=\{y\}\), any singleton.
Solution
1. (a) \([0,1]\) deformation retracts onto \(\{0\}\), so it also retracts onto \(\{0\}\).
\(H(x, t) = (1-t)x\) is continuous with \(H(x, 0) = x\), \(H(x, 1) = 0\), and \(H(0, t) = 0\) for all \(t\); its end map \(r(x) = 0\) is a retraction.
2. (b) \([0,1]\) admits no retraction onto \(\{0,1\}\), hence no deformation retraction.
A retraction \(r\colon [0,1] \to \{0,1\}\) would be a continuous surjection from a connected space onto a disconnected space, contradicting that continuous images of connected spaces are connected. The end map of a deformation retraction is a retraction.
3. (c) \(S^1\) retracts onto \(\{y\}\) but does not deformation retract onto it.
The constant map \(r(x) = y\) is continuous and fixes \(y\), so it is a retraction. A deformation retraction onto \(\{y\}\) would make the inclusion \(\{y\} \hookrightarrow S^1\) a homotopy equivalence, but \(\pi_1(S^1) \cong \ZZ\) while \(\pi_1(\{y\}) = 0\).