2.2 The Topology of a Metric Space

Since we will want to consider the properties of continuous functions in settings other than the Real Line, we review the material we just covered in the more general setting of Metric Spaces.

2.2.1 Definition:

A Metric Space, MATH is a set $\QTR{Large}{M}$ and a function MATH.

such that for all MATH $\QTR{Large}{M}$:

  1. Positivity: If MATH and MATH $\QTR{Large}{.}$

  2. Symmetry: MATH

  3. Triangle Inequality: MATH

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2.2.2 Lemma:

The Triangle Inequality has a second, equivalent form:

For all MATH

Proof:

Note that both formulas are symetric in MATH and MATH thus we may assume that

MATH . Thus can restate the formula in the statement of 2.2.2 as

MATH MATH

Adding MATH to or subtracting it from one inequality or the other gives the desired equivalence.

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2.2.3 Examples:

  1. For MATH MATH where, for example, MATH

  2. For MATH MATH

    more generally, for MATH MATH

  3. For MATH MATH

Note: For MATH

MATH

Note: For $\QTR{Large}{n=1\ }$the three metrics agree.

Exercise:

For MATH

  1. MATH

  2. MATH

Caculate MATH and MATH .

In fact the metrics generate the same "Topology" in a sense that will be made precise below.

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Examples 2.2.4:

  1. For any Metric Space MATH MATHis also a metric space. The base is not important.

    MATH

    MATH

  2. We will also want to understand the topology of the circle, MATH

There are three metrics illustrated in the diagram.

  1. MATH

  2. MATH

  3. MATH

Unlike the metrics for MATH , there is no simple function relationship between these metrics. However, they also all generate the same "Topology" on MATH

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2.2.5 Definition:

Fix a Metric Space $\QTR{Large}{(M,d)}$ and a point MATH

  1. Given MATH we define the open ball of radius MATH around $\QTR{Large}{x,}$ MATH MATH

  2. We say MATH is open if for any MATH there exist an MATH such that MATH

  3. The set of open sets is called the Topology defined by the Metric.

2.2.6 Theorem:

For any MATH is open .

Proof:

For every MATH . We need to find some MATH such that MATHLet MATH. Choose MATHFor every MATH . We have

MATH.

Thus MATH .

2.2.7 Theorem:

In a Metric Space $\QTR{Large}{(M,d)}$

  1. $\QTR{Large}{M}$ and MATH (the empty set) is open

  2. Let MATH be an arbitrary set of open sets, then MATH is also open.

  3. Let MATH be a finite set of open sets, then MATH is also open.

Proof:

1. and 2. follow directly from the definitions, however if you haven't been through this material you should write down the details.

3. makes use of the observation that if MATH then MATH. Choose MATH . Let

MATH . We have MATH for all $\QTR{Large}{i}$ hence MATH.

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2.2.8 Definition:

Given Metric Spaces MATH ,MATH and a function MATH. We say that $\QTR{Large}{f}$ is continuous a point $\QTR{Large}{x\in }$ MATH if given any MATH there is a MATH such thatMATH.

We say that $\QTR{Large}{f}$ is continuous if it is continuous at every point.

Exercise: Convince yourself that for MATH this is just the usual MATH definition.

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2.2.9 Theorem:

Given Metric Spaces MATH and MATH , a function MATH is continuous if and only for every open set MATH

MATH is open in MATH

Proof:

MATH

Suppose for every open set MATH we know that MATH is open in MATH Suppose we are given MATH and $\QTR{Large}{x\in }$ MATH. We need to find MATH such that MATH. But MATH is open. Hence MATH is open. Hence we can find MATH such that MATH . Or, equivalently, MATH

MATH

Suppose MATH is continuous and MATH is open we need to show that MATH is open in MATHSuppose we are given MATHand MATH MATH such that MATHSince $\QTR{Large}{f}$ is continuous, we can find MATH such that $\ $

MATH Thus MATH

2.2.10 Corollary

Continuity is determined by the underlying Topology. That is, if two Metrics define the same open sets then functions are continuous with respect to the first metric if and only if they are continuous with respect to the second.

2.2.11 Corollary

Suppose we have three Metric Spaces MATH ,MATH , and MATH suppose MATH and MATH are continuous, then so is MATH

Proof:

Let MATH be open. Since $\QTR{Large}{g}$ is continuous so is MATH is open in MATH and since $\QTR{Large}{f}$ is continuous

so is MATH

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Returning to the example of the tree metrics on the circle, MATH we want to show that the three Metrics

  1. MATH

  2. MATH

  3. MATH

produce the same set of continuous functions.

Solution:

We need to show that MATH is open with respect to MATH iff it is open with respect to MATHiff it is open with respect to MATH

To accomplish this, it suffices to show that if we are given $\QTR{Large}{x\in }$ MATHand MATH we can find, in sequence MATH , MATH and MATHsuch that

MATH

One note: there is no loss of generality in assuming that the MATHor MATH we work with for the next computation are smaller than the one we initially are given, or compute.

Beginning with a simple case, let MATHSuppose we know that MATH then we know MATH

So MATH .

The hardest calculations involve MATH . Given MATH , I need find MATH

$\vspace{1pt}$

MATH

In computing MATH we make use of the inequality MATH for small positive values of MATH.

Let MATH

then MATH and MATH for small MATH

Now let MATH . Suppose MATH

we have MATH or MATH

The computation of MATH follows in a similar fashion using the observation that MATH for MATH $\QTR{Large}{0}$.