Open Sets and Closed Sets in Metric Spaces

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Open and Closed Balls in Metric Spaces

Recall from the <a href="/open-and-closed-balls-in-euclidean-space">Open and Closed Balls in Euclidean Space</a> page that if then the open ball centered at with radius is as the set of points contained in:

Similarly, the closed ball centered at with radius is the set of points contained in:

Now notice that is simply the Euclidean distance function for the metric space . We can extend the concept of open and closed balls to any metric space with its own defined metric as defined below.

Definition: If is a metric space, , and then the Open Ball centered at with radius is defined to be the set .

For example, consider the metric space where the metric for all by:

Then for each , we can define the open ball centered at and with radius to be the set of all points contained in:

We can likewise define the closed ball centered at for some metric space as follows.

Definition: If is a metric space, , and then the Closed Ball centered at with radius is defined to be the set .

Note the subtle but important difference between the definitions of open and closed balls centered at with radius .

From the example above, we can define the closed ball centered at and with radius to be the set of all points contained in:


Open and Closed Sets in Metric Spaces

A set is said to be open if , that is, for every point we have that there exists a positive real number such that the ball centered at with radius is contained in , i.e., .

Furthermore, we said that is closed if is open.

For any general metric space , we define open and closeds subsets of in a similar manner.

Definition: If is a metric space and then is said to be Open if and is said to be Closed if is open. Moreover, is said to be Clopen if it is both open and closed.

It is important to note that the definitions above are somewhat of a poor choice of words. A set may just be open, just closed, open and closed (clopen), or even neither. Unfortunately these definitions are standard and we should note that saying a set is "not open" does not mean it is closed and likewise, saying a set is "not closed" does not mean it is open.

Now consider the whole set . Is open or closed? Well by definition, for every there exists a positive real number such that since the ball centered at with radius is defined to be the set of all points IN that are of a distance less than of . Therefore is an open set.

So then the complement of is is a closed set. However, it is vacuously true that for all there exists a ball centered at fully contained in since contains no points to begin with. Therefore is also an open set and so is also a closed set.

This is the case for all metric spaces . The whole set and empty set are trivially clopen sets!

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