EDB — 0B2

[0B2] The deleted neighbourhoods (sometimes called punctured neighbourhoods) of points \(x_ 0∈ℝ\) are divided into three classes.

• Neighborhoods of \(x_ 0∈ℝ\), which contain a set of the type \((x_ 0-𝛿,x_ 0)∪(x_ 0,x_ 0+𝛿)\) for a \(𝛿{\gt}0\);

• right neighborhoods of \(x_ 0∈ℝ\) , which contain a set of the type \((x_ 0,x_ 0+𝛿)\) for a \(𝛿{\gt}0\);

• left neighborhoods of \(x_ 0∈ℝ\) , which contain a set of the type \((x_ 0-𝛿,x_ 0)\) for a \(𝛿{\gt}0\);

In any case, the deleted neighborhoods must not contain the point \(x_ 0\). The ”full” neighborhoods are obtained by adding \(x_ 0\). The ”full neighborhoods” are the base for the standard topology on \(ℝ\).

To the previous ones we then add the neighborhoods of \(±∞\):

• neighborhoods of \(∞\) , which contain a set of the type \((y,∞)\) as \(y∈ℝ\) varies;

• neighborhoods of \(-∞\) , which contain a set of the type \((-∞,y)\) as \(y∈ℝ\) varies;

In this case we do not distinguish "deleted" neighborhoods and "full" neighborhoods.