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 ?