Open and closed sets of metric space

[sampahmel]

Homework Statement

I am using Rosenlicht's Intro to Analysis to self-study.

1.) I learn that the complements of an open ball is a closed ball. And...
2.) Some subsets of metric space are neither open nor closed.

Homework Equations

Is something amiss here? I do not understand how both can be true at the same time.

 

[HallsofIvy]

Homework Statement

I am using Rosenlicht's Intro to Analysis to self-study.

1.) I learn that the complements of an open ball is a closed ball.

No. The complement of an open set is an closed set but the complement of a "ball" is not a "ball". An open ball is of the form B_r(p)= \{ q| d(p, q)< r\right}. In R, an "open ball" is an open interval, (a, b). Its complement is (-\infty, a]\cup [b, \infty) which is closed but not a "ball".

And...
2.) Some subsets of metric space are neither open nor closed.

Homework Equations

Is something amiss here? I do not understand how both can be true at the same time.

I don't see what one has to do with the other. The complement of any open set is closed, the complement of any closed set is open. The complement of a set that is neither closed nor open is neither closed nor open. The "half open interval" in R, (0, 1], is neither closed nor open.

By the way, there also exists sets in a metric space that are both open and closed!

 

[sampahmel]

But I know that in any metric space, an open ball is an open set/ closed ball is a close set. Also, the complement of an open set is a closed set.

But then according to you,

The complement of an open ball is not closed ball.

So an open set is not an open ball?

 

[jbunniii]

There are open sets that are not open balls. For example, a set consisting of the union of two disjoint open balls is an open set, but it is not an open ball.

An open ball is a set consisting of all points less than a certain distance from a given point.

An open set is any set with the following property: no point is so close to the "boundary" that I can't center a suitably small open ball around that point, such that the ball is entirely contained in the set.

Every open ball is an open set but not vice versa.