EDB — 0HR

[0HR] Prerequisites:[0KX]↺↻,[0KZ]↺↻.Let $X=ℝ\cup \{\infty \}$, let’s consider the family $\mathcal{B}$ of parts of $X$ comprised of

• the open intervals $(a,b)$ with $a,b\in ℝ$ and $a<b$,

• the sets $(a,+\infty )\cup (-\infty ,b)\cup \{\infty \}$ with $a,b\in ℝ$ and $a<b$.

Show that $\mathcal{B}$ satisfies the properties (a),(b) seen in [0KX]↺↻. Let $𝜏$ therefore be the topology generated by this base. The topological space $(X,𝜏)$ is called one-point compactified line. This topological space is $T_2$ and it is compact (Exer. [0JD]↺↻); it is homeomorphic to the circle (Exer. [0YF]↺↻); therefore it can be equipped with a distance that generates the topology described above.