EDB — 0HR

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

• the open intervals \((a,b)\) with \(a,b∈ℝ\) and \(a{\lt}b\),

• the sets \((a,+∞)∪(-∞,b)∪\{ ∞\} \) with \(a,b∈ℝ\) and \(a{\lt}b\).

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.