Exercise 29.8

Proof. Let $K=\{1∕n\mid n\in {\mathbb{Z}}_{\text{+}}\}$. Let $f:{ℝ}_{\text{+}}\to {ℝ}_{\text{+}}$ such that $f(x)=1∕x$; this is a homeomorphism since it is continuous and is its own inverse. By Theorem $18.2(d)$ and $18.2(e)$, $f:{\mathbb{Z}}_{\text{+}}\to f({\mathbb{Z}}_{\text{+}})=K$ is continuous, and again is a homeomorphism since it is its own inverse. Now consider $Y=\{0\}\cup K$, which is closed and bounded and therefore compact by Theorem $27.3$, and Hausdorff by Theorem $17.11$. Since $K^{\prime }=\{0\}$ by Example 17.8, we know $Y$ is the one-point compactification of $K$. If $X=\{p\}\cup {\mathbb{Z}}_{\text{+}}$ is the one-point compactification of ${\mathbb{Z}}_{\text{+}}$, and letting $g:p\mapsto 0\in Y$, which is clearly continuous, the function $h:X\to Y$ defined by the pasting lemma (Theorem $18.3$) applied to $f,g$ is also continuous, and has continuous inverse defined by the pasting lemma applied to $f^{-1},g^{-1}$, and so is a homeomorphism $X\leftrightarrow Y$. □