Verify the statements made in Example 4 of §38 concerning the one-point compactification of \(X = (0, 1)\) obtained from the imbedding \(h: (0, 1) \to \mathbb{R}^2\) with component functions \(x\) and \(\sin(1/x)\).
Solution
Goal: Verify the topological properties of the compactification of \(X = (0, 1)\) formed by the closure of the embedding \(h(x) = (x, \sin(1/x))\) in \(\mathbb{R}^2\), its remainder structure, and its relationship to the one-point compactification.
1. Embedding and closure in \(\mathbb{R}^2\):Proof:
1.1. The map \(h: (0, 1) \to \mathbb{R}^2\) defined by \(h(x) = (x, \sin(1/x))\) is continuous and injective.
1.2. The inverse projection \(\pi_1: h(X) \to (0, 1)\) is continuous, so \(h\) is a topological embedding onto the planar subspace \(S = h((0, 1))\).
1.3. The closure \(Y = \overline{S}\) in \(\mathbb{R}^2\) is:\[Y = S \cup (\{0\} \times [-1, 1]) \cup \{(1, \sin 1)\}.\]
1.4. The space \(Y\) is closed and bounded in \(\mathbb{R}^2\), hence compact by the Heine-Borel theorem. Thus \(Y\) is a compactification of \((0, 1)\).
2. Structure of the remainder and non-extendability:Proof:
2.1. The remainder \(Y \setminus S = (\{0\} \times [-1, 1]) \cup \{(1, \sin 1)\}\) contains an entire continuum of limit points \(\{0\} \times [-1, 1]\) corresponding to the end \(x \to 0^+\).
2.2. For any \(y \in [-1, 1]\), the sequence \(x_n = \frac{1}{\arcsin(y) + 2\pi n} \to 0^+\) satisfies \(h(x_n) \to (0, y)\).
2.3. Consequently, \(\lim_{x \to 0^+} h(x)\) does not exist in \(\mathbb{R}^2\).
2.4. Therefore, the embedding \(h\) cannot be extended continuously to the one-point compactification \(X^* = (0, 1) \cup \{\infty\} \cong S^1\).
3. Quotient relationship to the one-point compactification:Proof:
3.1. The one-point compactification of \((0, 1)\) is the circle \(S^1\), obtained by adding a single point \(\infty\) with open neighborhoods being complements of compact subsets of \((0, 1)\).
3.2. If one forms the quotient space \(Y / (Y \setminus S)\) by collapsing the entire compact remainder \((\{0\} \times [-1, 1]) \cup \{(1, \sin 1)\}\) to a single point, the resulting quotient space is homeomorphic to the one-point compactification \(X^* \cong S^1\).
3.3. This illustrates that any compactification of a locally compact Hausdorff space \(X\) projects onto the one-point compactification \(X^*\) by collapsing the remainder to a point. Q.E.D.