Hacker News Reader

Comment by Nitrolo | Hacker News Reader

Here's an interesting one: "this set is open, hence it is not closed". Apparently this is wrong, and R is both open and closed. Does anyone have a clear explanation as to why this is or what this means? I can't quite understand the comment explaining it.

View on HN

Let me try to explain it: One usually thinks of open sets as collections of numbers without a "boundary" - the boundary of the set is not part of the set but outside of it. If we look at the open set, we see that it's just "interior" points. You may have seen this when distinguishing on the number line between intervals like (-1,2) and [-1,2].

(-1,2) are all number between -1 and 2, but not including -1 and 2. [-1,2] however are the same numbers, but including -1 and 2. Slightly different sets, where the former is open and the latter is closed.

Another way to think about it is for any point in (-1,2), we can find some number that is "closer to the boundary" but still in the set. This is not true for [-1,2] if we pick -1, because any number below -1 is suddenly outside the set. -1 is a boundary point, and therefore that side of the interval is not open!

Why do we need this? Well, lots of calculus is about sequences that converge in smaller and smaller distances. One can for example define whether a function "jumps" by checking whether the inputs have boundary points or not (etc.).

But for math people, this is not general enough, because it requires things like distances. Hence, they have a more general definition: There's a collection of stuff, and open sets are sets of this stuff that follow some rules.

Now here is the important part: Closed sets do not have an extra set of rules. Instead, they are just defined as the complement (or opposite) of an open set.

So R - the set of all real numbers - has a complement which is the empty set. The opposite of "all numbers" is nothing! Since R is open, by definition, the empty set is closed.

Cool. But then, if we take our rules, we see the the empty set is also open. Whoops. Then, again by definition - the opposite of the empty set - which is R, so all numbers - is closed.

Confusion conclusion: Both R and its complement, the empty set, are both closed and open at the same time.

We just call those sets "clopen".

Thank you for the clear explanation! Pretty sure we talked about open and closed intervals in my first week of undergrad but as a mechanical engineer we stopped there and never took a look at the more general concept.

You can define closed sets positively as well. A closed set is one which includes its whole boundary. An open set is one which excludes its whole boundary. Clearly then it is possible for a set to contain only part of its boundary, in which case it's neither closed nor open. Somewhat less obviously, it's possible for a set's boundary to be empty, in which case it is both closed and open.

It's just unfortunate terminology -- a closed set is one whose complement is open, not one which is not open.

Indeed. Sets are not doors, as the saying goes, because they can be open and closed at the same time.

It's not a common false belief, IMO. On the first day you learn about open and closed sets, you learn that the for any metric space S, the set S is open in S and the empty set is open in S, and they're closed as well. (Students typically get introduced the concepts in the context of metric spaces.)

Note that sets are always closed or open relative to some other specified (often implied) set S.

In some courses, a closed set (in S) is defined to be a set which is the complement S-T for some open set T. In others, a closed set (in S) is defined to be a set which contains all its limit points (in S). And then whichever isn't the definition gets proven as a theorem.

I think the first _and_only_ day many people learn about open and closed sets is when they learn about the specific example of open and closed intervals in ℝ. That happens early in high school.

The misconception is about not knowing that the generalization to arbitrary sets has some unexpected properties.

(Half-open intervals are neither open nor closed, by the way)

I never learned about open and closed "sets" in R, in high school, only intervals.

I guess all intervals are sets (they are a collection of numbers after all) but not all sets are intervals. Any mathematicians here to tell me if I just said something stupid?

No, that's correct.

Indeed, sets come later, around university, and I would assume that most university fields that use mathematics do NOT need to care about some sets like R (and C ?) having the clopen property ?

So can a set that's not an interval be open or closed?

Reply on news.ycombinator.com