44 CHAPTER 2. BASIC TOPOLOGY AND ALGEBRA

As for the last claim, let Q be the rational numbers. Then obviously Qp is dense in Rp

and (Q+ iQ)p is dense in Cp. There are countably many points in (Q+ iQ)p by inductionapplied to Theorem 1.2.7. ■

Definition 2.4.3 Let S be a nonempty set. Then a set of open sets C is called anopen cover of S if ∪C ⊇S . (It covers up the set S. Think lilly pads covering the surfaceof a pond.)

One of the important properties possessed by separable metric spaces is the Lindeloffproperty.

Definition 2.4.4 A metric space has the Lindeloff property if whenever C is an opencover of a set S, there exists a countable subset of C denoted here by B such that B is alsoan open cover of S.

Theorem 2.4.5 Every separable metric space has the Lindeloff property.

Proof: Let C be an open cover of a set S. Let B be a countable basis. Such exists byTheorem 2.4.2. Let B̂ denote those sets of B which are contained in some set of C . ThusB̂ is a countable open cover of S. Now for B ∈ B̂, let UB be a set of C which contains B.Letting Ĉ denote these sets UB it follows that Ĉ is countable and is an open cover of S. ■

Note how the axiom of choice was used in the above where we let UB be a set of Cwhich contains B.

Definition 2.4.6 A Polish space is a complete separable metric space. These thingsturn out to be very useful in probability theory and in other areas.

Now it is convenient to consider the distance function in a metric space (X ,d).

Definition 2.4.7 Let S be a nonempty set in X and let x ∈ X . Then the distance of xto the set S is defined as

dist(x,S)≡ inf{d (x,y) : y ∈ S}

The main result concerning this function is that it is Lipschitz continuous as describedin the following theorem.

Theorem 2.4.8 Let S ̸= /0 and consider f (x)≡ dist(x,S) , then

| f (x)− f (x̂)| ≤ d (x, x̂) .

Proof: Say dist(x,S) ≤ dist(x̂,S). Otherwise, reverse the argument which follows.Then for a suitable choice of y ∈ S,

|dist(x,S)−dist(x̂,S)|= dist(x̂,S)−dist(x,S)≤ dist(x̂,S)− (d (x,y)− ε)

Then|dist(x,S)−dist(x̂,S)| ≤ d (x̂,y)− (d (x,y)− ε)

≤ d (x̂,x)+d (x,y)−d (x,y)+ ε = d (x̂,x)+ ε

Since ε is arbitrary, this shows the claimed result. ■