EDB — 0HP

[0HP] Prerequisites:[0KX]↺↻,[0KZ]↺↻.Let \(X=ℝ∪\{ +∞,-∞\} \), consider the family \(\mathcal B\) of parts of \(X\) that contains

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

• half-lines \((a,+∞]=(a,+∞)∪\{ +∞\} \) with \(a∈ℝ\),

• the half-lines \([-∞,b)=(-∞,b)∪\{ -∞\} \) with \(b∈ℝ\).

(Note the similarity of sets in the second and third points with the ”neighbourhoods of infinity” seen in Sec. [29H]↺↻).

Show that \(\mathcal B\) satisfies the properties (a),(b) seen in [0KX]↺↻. Let \(𝜏\) therefore be the topology generated from this base. The topological space \((X,𝜏)\) is called extended line, often denoted \(\overlineℝ\).

This topological space is \(T_ 2\), it is compact (Exercise [0JB]↺↻), and is homoemorphic to the interval \([0,1]\). It can be equipped with a distance that generates the topology described above.