# Note 7 - Metric Spaces

## Table of Contents

## Metric Space

- Simplest mathematical structure. We need to talk about “distance,” which allows us to talk about “-neighborhoods” and “convergence”, etc.

**Definition**: A **metric space** is a set , and a distance function that satisfies:

### Examples

## Neighborhoods

**Definition**: Let be a metric space, . We can create a **Ball** with radius that contains all elements within that radius, denoted .

### Examples

.

circle with radius 1, boundary excluded.

## Open Subsets

**Definition**: Let be a metric space, and . We say is an **open subset** of if such that .

### Examples

**Q:** Let be a metric space. Is an open subset?

Answer: No. If we take a look at the “edge” (ie. point 0), we can see that the ball around this point is not in the set. Formally, since implies the range , which contains elements outside.

**Q:** Taking the same setup above, is open?

Answer: Any element must be in , so we can always take the radius to be half the distance to the shorter border.

Formally, if , then .

**Q:** In general, for any metric space, is open?

To prove this, we must show , we have some such that .

Let us define .

Claim: .

. By Triangle Inequality, . This means that , which completes the proof.

## Closed Subsets

**Definition**: Let be a metric space with . We say is a **closed subset** of if is open (where .

### Examples

**Q:** Is closed?

Yes. . We can then use the same argument as above to show that it is open.

**Q:** Is closed?

No. . We can then use the same argument as above to show that it is not open.

**Q:** Is closed?

No. . For any radius around , we can find a such that it’s within this radius.

Equivalently, always contains some elements in . This implies that , so is not closed.

These examples illustrate that *for a set to be closed, it should contain all its limit points*.

## Limit Points

**Definition**: Let be a metric space, with . We call a **limit point** of if contains a point such that .

### Examples

**Q:** What are the limit points of ?

is a limit point, since there exists points in Every point in is a limit point of .

**Q:** Is it true that *any point* is a limit point of ?

No. Take the example . is not a limit point of .

## Closure

**Definition**: The **closure** of is defined to be the union of and its limit points.

e.g.

**Theorem**: Let be a metric space, and . We claim that is closed if and only if contains all the limit points of .

Proof : We want to show that if , then is not a limit point of .

if and only if is open. This implies that such that , which means that . This shows that x is not a limit point of .Proof : We want to show is open. , we know , which means is not a limit point of . This means such that is either \emptyset or . The second case cannot be true, since . Therefore, , so . This shows that is open, so is closed.

**Exercise**: Is open or closed? What about ?