EDB — 2C2

10.2 Topology in metric spaces[2C2]

Let \((X,d)\) be a metric space.

Note that, having the operational definition [(9.22)]↺↻ of ”open set”, then the axioms (in the definition [0G6]↺↻) in this case become theorems.

E20

[0NZ]↺↻

E20

[0P1]↺↻

E20

[0P3]↺↻

E20

[0P5]↺↻

E20

[0P6]↺↻

E20

[0P8]↺↻

E20

[0PB]↺↻

E20

[0PD]↺↻

E20

[0PG]↺↻

E20

[0PJ]↺↻

E20

[0PM]↺↻

E20

[0PP]↺↻

E20

[0PQ]↺↻

E20

[0PR]↺↻

E20

[0PS]↺↻

E20

[0PT]↺↻

E20

[0PY]↺↻

E20

[0Q0]↺↻

E20

[0Q3]↺↻

E20

[0Q5]↺↻

E20

[0Q7]↺↻

E20

[0Q8]↺↻

E20

[0QC]↺↻

E20

[0QF]↺↻

Bases composed of balls

To face these exercises it is necessary to know the concepts seen in Sec. [2B5]↺↻.

E20

[0QJ]↺↻

E20

[0QM]↺↻

Accumulation points, limit points

Let’s redefine this notion (a special case of what we saw in [0GY]↺↻)

The set of accumulation points of \(A\) is called derived set, we will indicate it with \(D(A)\).

E21

[0QP]↺↻

E21

[0QR]↺↻

E21

[0QS]↺↻

E21

[0QV]↺↻

Let’s add this definition (a special case of [2B4]↺↻).

In English literature the terms ”cluster point”, ”limit point” and ”accumulation point” are sometimes considered synonimous, which can be confusing. We will stick to the proposed definitions [0QN]↺↻ and [0QX]↺↻.

E22

[0QY]↺↻

E22

[0QZ]↺↻

E22

[2F3]↺↻

Other exercises on these topics are [0S8]↺↻, [0SB]↺↻, [0SD]↺↻, [0SN]↺↻ and [0T5]↺↻.