Show that the one-point compactification of \(\mathbb{Z}_+\) is homeomorphic with the subspace \(\ts{0} \cup \ts{1/n \mid n \in \mathbb{Z}_+}\) of \(\mathbb{R}\).
Solution
Goal: Prove that the one-point compactification \(\mathbb{Z}_+^* = \mathbb{Z}_+ \cup \{\infty\}\) of the discrete space \(\mathbb{Z}_+\) is homeomorphic to the subspace \(K = \{0\} \cup \{1/n \mid n \in \mathbb{Z}_+\} \subset \mathbb{R}\).
1. Structure of the one-point compactification \(\mathbb{Z}_+^*\):Proof:
1.1. The positive integers \(\mathbb{Z}_+\) carry the discrete topology, where every subset is open and compact subsets are precisely the finite subsets.
1.2. The one-point compactification \(\mathbb{Z}_+^* = \mathbb{Z}_+ \cup \{\infty\}\) is a compact Hausdorff space whose topology consists of all subsets of \(\mathbb{Z}_+\) together with all sets of the form \((\mathbb{Z}_+ \setminus F) \cup \{\infty\}\) where \(F \subset \mathbb{Z}_+\) is finite.
2. Construction and bijectivity of the candidate map:Define \(f: \mathbb{Z}_+^* \to K\) by:\[f(n) = \frac{1}{n} \quad \text{for } n \in \mathbb{Z}_+, \qquad f(\infty) = 0.\]Proof:
2.1. \(f\) maps \(\mathbb{Z}_+\) bijectively onto \(\{1/n \mid n \in \mathbb{Z}_+\}\).
2.2. \(f\) maps \(\infty\) to \(0 \notin \{1/n \mid n \in \mathbb{Z}_+\}\).
2.3. Thus \(f: \mathbb{Z}_+^* \to K\) is a well-defined bijection.
3. Continuity of \(f\):Proof:
3.1. For each \(n \in \mathbb{Z}_+\), \(\{n\}\) is an open singleton in \(\mathbb{Z}_+^*\), so \(f\) is continuous at \(n\).
3.2. Let \(V \subseteq K\) be an open neighborhood of \(f(\infty) = 0\) in the subspace topology of \(K\).
3.3. There exists \(\varepsilon > 0\) such that \((-\varepsilon, \varepsilon) \cap K \subseteq V\).
3.4. Choose \(N \in \mathbb{Z}_+\) such that \(\frac{1}{N} < \varepsilon\).
3.5. For all \(n \ge N\), \(0 < f(n) = \frac{1}{n} \le \frac{1}{N} < \varepsilon\), so \(f(n) \in V\).
3.6. Thus \(f^{-1}(V) \supseteq (\mathbb{Z}_+ \setminus \{1, \dots, N-1\}) \cup \{\infty\}\).
3.7. Because \(\{1, \dots, N-1\}\) is finite, this preimage is an open neighborhood of \(\infty\) in \(\mathbb{Z}_+^*\).
3.8. Hence \(f\) is continuous at \(\infty\), and thus continuous on all of \(\mathbb{Z}_+^*\).
4. Homeomorphism conclusion:Proof:
4.1. The domain \(\mathbb{Z}_+^*\) is compact.
4.2. The codomain \(K \subset \mathbb{R}\) is Hausdorff.
4.3. Any continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
4.4. Therefore, \(f: \mathbb{Z}_+^* \to K\) is a homeomorphism. Q.E.D.