6.1 Neighbourhoods[29H]
(Solved on 2022-11-24)
Neighbourhoods are a family of sets associated with a point \(x_ 0∈ℝ\), or \(x_ 0=±∞\). The neighbourhoods are sets that contain an ”example” set. Let’s see here some definitions.
[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.
[29J]Prerequisites:53.Difficulty:*.(Proposed on 2022-11-24) Let \(x_ 0\in \overline{{\mathbb {R}}}\) and \(\mathcal F\) all the neighbourhoods of \(x_ 0\). We associate the ordering
show that this is a filtering ordering.
(This holds both for “deleted” and for “full” neighbourhoods; for ‘left”, “right”, or “bilateral” neighbourhoods).
(See also 8 for the similar statement in topological spaces).