Let \(X\) be the adjunction space formed from the disjoint union of the normal, path-connected space \(A\) and the unit ball \(B^2\) by means of a continuous map \(f: S^1 \to A\). (See E-HYC1G.) Show that \(X\) satisfies the hypotheses of Theorem 72.1. Where do you use the fact that \(A\) is normal?
Solution
Goal: Verify that the adjunction space \(X = A \cup_f B^2\) satisfies the three structural hypotheses of Theorem 72.1 for cell attachment, and identify precisely where normality of \(A\) is needed.
1. Hypotheses of Theorem 72.1:A space \(X\) obtained by attaching a 2-cell to a subspace \(A\) must satisfy:1. \(A\) is a closed subspace of \(X\), and \(X\) is Hausdorff.2. \(X \setminus A\) is homeomorphic to the open 2-cell \(\operatorname{Int} B^2 = \mathbb{B}^2\).3. There exists an open neighborhood \(U\) of \(A\) in \(X\) that deformation retracts onto \(A\).
2. Verification of Hypothesis 1 (\(A\) is closed and \(X\) is Hausdorff):Proof:
2.1. Let \(\pi: A \sqcup B^2 \to X\) be the quotient projection.
2.2. \(\pi^{-1}(A) = A \sqcup S^1\), which is closed in \(A \sqcup B^2\). By the quotient topology, \(A\) is a closed subspace of \(X\), and \(\pi|_A: A \to \pi(A) \cong A\) is a topological embedding.
2.3. Use of Normality: To prove \(X\) is Hausdorff (and normal):- Disjoint closed sets in \(A\) can be separated by Urysohn functions because \(A\) is normal (\(T_4\)).- Since \(B^2\) is compact Hausdorff (hence normal) and the attaching region \(S^1 \subset B^2\) is closed, the adjunction space \(X = A \cup_f B^2\) of a normal space \(A\) along a compact metric space \(B^2\) is normal, hence Hausdorff (\(T_2\)).- (If \(A\) were not normal, the quotient \(A \cup_f B^2\) could fail to be Hausdorff at boundary points of \(f(S^1)\)).
3. Verification of Hypothesis 2 (\(X \setminus A\) is an open 2-disk):Proof:
3.1. The restriction of \(\pi\) to the open ball \(\operatorname{Int} B^2\) is an injective continuous map onto \(X \setminus A\).
3.2. Since \(\operatorname{Int} B^2\) is saturated under \(\pi\) (no points in \(\operatorname{Int} B^2\) are identified with other points), \(\pi|_{\operatorname{Int} B^2}: \operatorname{Int} B^2 \to X \setminus A\) is a homeomorphism.
4. Verification of Hypothesis 3 (Deformation retract neighborhood \(U\)):Proof:
4.1. Let \(U = X \setminus \{\pi(0)\}\). Then \(\pi^{-1}(U) = A \sqcup (B^2 \setminus \{0\})\), which is open in \(A \sqcup B^2\), so \(U\) is an open neighborhood of \(A\) in \(X\).
4.2. In \(B^2 \setminus \{0\}\), define the standard straight-line deformation retraction onto \(S^1\):\[r_t(x) = \left((1-t) + t \frac{1}{\|x\|}\right) x \quad \text{for } x \in B^2 \setminus \{0\}, \; t \in [0, 1].\]
4.3. Define \(H_t: A \sqcup (B^2 \setminus \{0\}) \to A \sqcup (B^2 \setminus \{0\})\) by \(H_t|_A = \operatorname{id}_A\) and \(H_t|_{B^2 \setminus \{0\}} = r_t\).
4.4. Since \(r_t|_{S^1} = \operatorname{id}_{S^1}\) for all \(t\), \(H_t\) respects the quotient identifications, inducing a continuous homotopy \(\bar{H}_t: U \to U\) with \(\bar{H}_0 = \operatorname{id}_U\) and \(\bar{H}_1(U) = A\).
4.5. Thus \(U\) deformation retracts onto \(A\).
5. Conclusion:\(X\) satisfies all hypotheses of Theorem 72.1. Normality of \(A\) is used specifically in 2 to guarantee that the adjunction space \(X\) is Hausdorff. Q.E.D.