Recall that a space \(X\) is said to be contractible if the identity map of \(X\) to itself is nulhomotopic. Show that \(X\) is contractible if and only if \(X\) has the homotopy type of a one-point space.
Solution
Goal: Prove that a topological space \(X\) is contractible (i.e. \(\operatorname{id}_X\) is nullhomotopic) if and only if \(X\) is homotopy equivalent to a one-point space \(P = \{p_0\}\).
1. Direct implication (\(\implies\)): If \(X\) is contractible, then \(X \simeq \{p_0\}\).Proof:
1.1. By definition of contractibility, \(\operatorname{id}_X\) is nullhomotopic: there exists a point \(x_0 \in X\) and a continuous homotopy \(H: X \times I \to X\) such that \(H(x, 0) = x\) and \(H(x, 1) = x_0\) for all \(x \in X\).
1.2. Let \(P = \{p_0\}\) be the one-point space.
1.3. Define continuous maps \(f: X \to P\) by \(f(x) = p_0\) for all \(x \in X\), and \(g: P \to X\) by \(g(p_0) = x_0\).
1.4. The composition \(f \circ g: P \to P\) is the identity map \(\operatorname{id}_P\).
1.5. The composition \(g \circ f: X \to X\) is the constant map \((g \circ f)(x) = x_0\).
1.6. The homotopy \(H\) provides \(g \circ f \simeq \operatorname{id}_X\).
1.7. Thus \(f\) and \(g\) are homotopy inverse equivalences, so \(X\) has the homotopy type of a one-point space.
2. Converse implication (\(\impliedby\)): If \(X \simeq \{p_0\}\), then \(X\) is contractible.Proof:
2.1. Suppose there exist continuous maps \(f: X \to P\) and \(g: P \to X\) such that \(g \circ f \simeq \operatorname{id}_X\).
2.2. Let \(x_0 = g(p_0) \in X\).
2.3. Since \(P = \{p_0\}\), the only possible value for \(f(x)\) is \(p_0\), which means:\[(g \circ f)(x) = g(f(x)) = g(p_0) = x_0 \quad \text{for all } x \in X.\]
2.4. Thus \(g \circ f\) is the constant map \(c_{x_0}: X \to X\).
2.5. Because \(g \circ f \simeq \operatorname{id}_X\), the identity map \(\operatorname{id}_X\) is homotopic to the constant map \(c_{x_0}\).
2.6. By definition, \(\operatorname{id}_X\) is nullhomotopic, so \(X\) is contractible.
3. Conclusion:\(X\) is contractible if and only if \(X\) has the homotopy type of a one-point space. Q.E.D.