Topology/Euclidean Spaces

Euclidean space is the space in which everyone is most familiar. In Euclidean k-space, the distance between any two points is

where k is the dimension of the Euclidean space. Since the Euclidean k-space as a metric on it, it is also a topological space.

Sequences

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Definition: A sequence of real numbers   is said to converge to the real number s provided for each   there exists a number   such that   implies  .

Definition: A sequence   of real numbers is called a Cauchy sequence if for each   there exists a number   such that        .

Lemma 1

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Convergent sequences are Cauchy Sequences.

Proof : Suppose that  .

Then,

 

Let  . Then   such that

 

Also:

 

so

 

Hence,   is a Cauchy sequence.

Theorem 1

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Convergent sequences are bounded.

Proof: Let   be a convergent sequence and let  . From the definition of convergence and letting  , we can find N   such that

 

From the triangle inequality;

 

Let  .

Then,

 

for all  . Thus   is a bounded sequence.

Theorem 2

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In a complete space, a sequence is a convergent sequence if and only if it is a Cauchy sequence.

Proof:

  Convergent sequences are Cauchy sequences. See Lemma 1.

  Consider a Cauchy sequence  . Since Cauchy sequences are bounded, the only thing to show is:

 

Let  . Since   is a Cauchy sequence,   such that

 

So,   for all  . This shows that   is and upper bound for   and hence   for all  . Also  is a lower bound for  . Therefore  . Now:

 

Since this holds for all  ,  . The opposite inequality always holds and now we have established the theorem.

Note: The preceding proof assumes that the image space is  . Without this assumption, we will need more machinery to prove this.

Definition

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It is possible to have more than one metric prescribed to an image space. It is often better to talk about metric spaces in a more general sense. A sequence   in a metric space   converges to s in S if  . A sequence is called Cauchy if for each   there exists an   such that:

 

.

The metric space   is called complete if every Cauchy sequence in   converges to some element in  .

Theorem 3

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Let   be a complete metric and   be a subspace of  . Then   is a complete metric space if and only if   is a closed subset of  .

Proof:   Suppose   is a closed subset of  . Let   be a Cauchy sequence in  .

Then   is also a Cauchy sequence in  . Since   is complete,   converges to a point   in  . However,   is a closed subset of   so   is also complete.

  Left as an exercise.



Exercises

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1. Let  . Let   where   and  . Show:

a.)   and   are metrics for  .

b.)   and   form a complete metric space.

2. Show that every open set in   is the disjoint union of a finite or infinite sequence of open intervals.

3. Complete the proof for theorem 3.

4.Consider: Let   and   be metric spaces.

a.) A mapping   is said to be a Lipschitz mapping provided that there is some non-negative number c called a Lipschitz constant for the mapping such that:

 

b.) A Lipschitz mapping   that has a Lipschitz constant less than 1 is called a contraction.

Suppose that   and   are both Lipschitz. Is the product of these functions Lipschitz?