A space \(X\) is said to be contractible if the identity map \(i_X: X \to X\) is nulhomotopic.
Show that \(I\) and \(\mathbb{R}\) are contractible.
Show that a contractible space is path connected.
Show that if \(Y\) is contractible, then for any \(X\), the set \([X, Y]\) has a single element.
Show that if \(X\) is contractible and \(Y\) is path connected, then \([X, Y]\) has a single element.
Solution
1. Part (a): \(I = [0, 1]\) and \(\mathbb{R}\) are contractible.Proof:
1.1. For \(I = [0, 1]\), define \(H: I \times I \to I\) by \(H(x, t) = (1 - t)x\). Then \(H\) is continuous, \(H(x, 0) = x = \operatorname{id}_I(x)\), and \(H(x, 1) = 0\). Thus \(\operatorname{id}_I \simeq c_0\).
1.2. For \(\mathbb{R}\), define \(H: \mathbb{R} \times I \to \mathbb{R}\) by \(H(x, t) = (1 - t)x\). Then \(H\) is continuous, \(H(x, 0) = x = \operatorname{id}_{\mathbb{R}}(x)\), and \(H(x, 1) = 0\). Thus \(\operatorname{id}_{\mathbb{R}} \simeq c_0\).
2. Part (b): Contractible spaces are path-connected.Proof:
2.1. Let \(X\) be contractible, so there exists a null-homotopy \(H: X \times I \to X\) with \(H(x, 0) = x\) and \(H(x, 1) = x_0\) for some fixed \(x_0 \in X\).
2.2. For any point \(x \in X\), the curve \(\gamma_x: I \to X\) given by \(\gamma_x(t) = H(x, t)\) is a continuous path in \(X\) from \(\gamma_x(0) = x\) to \(\gamma_x(1) = x_0\).
2.3. For any two points \(a, b \in X\), the path product \(\gamma_a * \overline{\gamma_b}\) is a continuous path in \(X\) from \(a\) to \(b\).
2.4. Thus \(X\) is path-connected.
3. Part (c): If \(Y\) is contractible, then \([X, Y]\) is a singleton for any space \(X\).Proof:
3.1. Let \(H: Y \times I \to Y\) be a homotopy with \(H(y, 0) = y\) and \(H(y, 1) = y_0\).
3.2. For any continuous map \(f: X \to Y\), define \(G: X \times I \to Y\) by \(G(x, t) = H(f(x), t)\).
3.3. As a composition of continuous maps, \(G\) is continuous; moreover \(G(x, 0) = H(f(x), 0) = f(x)\) and \(G(x, 1) = H(f(x), 1) = y_0\).
3.4. Thus every map \(f: X \to Y\) is homotopic to the constant map \(c_{y_0}: X \to Y\).
3.5. Therefore, all maps in \(\mathcal{C}(X, Y)\) lie in the single homotopy class \([c_{y_0}]\), so \(|[X, Y]| = 1\).
4. Part (d): If \(X\) is contractible and \(Y\) is path-connected, then \([X, Y]\) is a singleton.Proof:
4.1. Let \(H: X \times I \to X\) be a null-homotopy with \(H(x, 0) = x\) and \(H(x, 1) = x_0\).
4.2. For any map \(f: X \to Y\), the homotopy \(F(x, t) = f(H(x, t))\) shows \(f \simeq c_{f(x_0)}\).
4.3. Since \(Y\) is path-connected, for any \(y_1, y_2 \in Y\) there exists a path \(\alpha: I \to Y\) with \(\alpha(0) = y_1\) and \(\alpha(1) = y_2\).
4.4. The homotopy \(K(x, t) = \alpha(t)\) establishes that \(c_{y_1} \simeq c_{y_2}\).
4.5. Consequently, for any two maps \(f, g: X \to Y\), \(f \simeq c_{f(x_0)} \simeq c_{g(x_0)} \simeq g\).
4.6. Thus all maps from \(X\) to \(Y\) are homotopic, so \(|[X, Y]| = 1\). Q.E.D.