5.1 Neighbourhoods[29H]

(Solved on 2022-11-24)

Neighbourhoods are a family of sets associated with a point x0∈ℝ, or x0=±∞. The neighbourhoods are sets that contain an ”example” set. Let’s see here some definitions.

Definition 173 Neighbourhoods

[0B2] The deleted neighbourhoods (sometimes called punctured neighbourhoods) of points x0∈ℝ are divided into three classes.

  • Neighborhoods of x0∈ℝ, which contain a set of the type (x0−𝛿,x0)∪(x0,x0+𝛿) for a 𝛿>0;

  • right neighborhoods of x0∈ℝ , which contain a set of the type (x0,x0+𝛿) for a 𝛿>0;

  • left neighborhoods of x0∈ℝ , which contain a set of the type (x0−𝛿,x0) for a 𝛿>0;

In any case, the deleted neighborhoods must not contain the point x0. The ”full” neighborhoods are obtained by adding x0. 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.

Exercise 174

[29J]Prerequisites:53.Difficulty:*.(Proposed on 2022-11-24) Let x0∈R― and F all the neighbourhoods of x0. We associate the ordering

I,J∈F  ,I≤J⟺I⊇J

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).