What is a clopen set and how can a set be both open and closed?

[ProfuselyQuarky]

I feel like I ask too many questions here, so I'm sorry about that. But, anyway, what is a clopen set? I was watching something the other day that a friend sent me, and, out of the blue, the guy starts referring to a set as being "clopen" with no explanation. I tried to break all the definitions I know down into little bits, but I still find it confusing. Apparently a clopen set is both open and closed. How?? Letting $A\subseteq \mathbb{R}$, I know that a set A is open if every point of A is an interior point of A . A is closed, however, if and only if $\mathbb{R}\setminus A$.

From Wiki:

A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open and closed, and therefore clopen.

I know that a complement of a set is all the things outside of the set, but I just don't understand. It's not the (singular) set itself that's both open and closed, right? All I get out from reading is that you can have a closed set with an open complement and an open set with an open complement. I can't see how that makes a set clopen. It is still either open or closed.

When it says "which leaves the possibility of an open set..." are they talking about a completely different set or something related to the initial closed set...?

Obviously I am wrong and lost, so clarification would be greatly appreciated. I am going in circles.

 

[jedishrfu]

From the wiki article they have this example:

As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, and the set A of all positive rational numbers whose square is bigger than 2. Using the fact that ${\displaystyle {\sqrt {2}}}$ is not in Q, one can show quite easily that A is a clopen subset of Q. (Note also that A is not a clopen subset of the real line R; it is neither open nor closed in R.)

 

[ProfuselyQuarky]

Well, I already read that. It doesn't really help though Another example that was given was:

In any topological space X, the empty set and the whole space X are both clopen.

I feel like this is something I should understand easily, but I'm not. I have to leave now, but will come back shortly, hopefully with different reasoning.

 

[The Bill]

Well, I already read that. It doesn't really help though Another example that was given was:

"In any topological space X, the empty set and the whole space X are both clopen."

I feel like this is something I should understand easily, but I'm not. I have to leave now, but will come back shortly, hopefully with different reasoning.

Recall that in the definition of a topological space (X,O), where O is the set of all sets defined to be open, both X and $\emptyset$ are defined to be open. That is, both X and $\emptyset$ are placed in O "by hand" in the very definition of a topological space.

Now, what is the complement of X? What is the complement of $\emptyset$? Are these complements open?

 

[ProfuselyQuarky]

Now, what is the complement of X? What is the complement of $\emptyset$? Are these complements open?

The complements are closed, right?

 

[The Bill]

The complements are closed, right?

In this specific case, X is closed because its complement $\emptyset$ is defined to be open, and $\emptyset$ is closed because its complement X is defined to be open. Therefore they're each both closed and open.

 

[ProfuselyQuarky]

In this specific case, X is closed because its complement $\emptyset$ is defined to be open, and $\emptyset$ is closed because its complement X is defined to be open. Therefore they're each both closed and open.

Ah! I think I understand now. This statement is ever so concise, thank you for the simple explanation.

As a side note, regarding the definition saying that "A set is open if its complement is closed", this obviously makes sense. However, I have seen a few (very few) places that say that "A set is open iff its complement is closed". This is an error, correct? I haven't read anything implying that the statement is supposed to be biconditional and the places that have mentioned it only noted it briefly without elaborating. If the biconditional version is correct, could you direct me to a proof for it?

 

[PeroK]

Ah! I think I understand now. This statement is ever so concise, thank you for the simple explanation.

As a side note, regarding the definition saying that "A set is open if its complement is closed", this obviously makes sense. However, I have seen a few (very few) places that say that "A set is open iff its complement is closed". This is an error, correct? I haven't read anything implying that the statement is supposed to be biconditional and the places that have mentioned it only noted it briefly without elaborating. If the biconditional version is correct, could you direct me to a proof for it?

In a definition "if" is often used instead of more precisely "iff". In this case, a set is closed iff its complement is open; which means a set is open iff its complement is closed.

 

[micromass]

It is a mistake to think that open and closed are two extremes. I understand that an open door and a closed door are as far apart as can be. But this situation is not true in mathematics anymore. I think the terminology open/closed is a very bad one, but it has unfortunately stuck.

There are many types of closedness in mathematics. But they all share the same characteristic. A set is closed if whatever you "do", you cannot go out of the set. Much like a prison. In topology, this translates to: if you are in the set and if you take infinitely many steps in the set, your position at infinity will still be in the set. More formally, if you have a sequence $(x_n)_n$ that converges to $x$, then if $x_n$ is in the set for each $n$, then so is $x$. (purists will no doubt tell me that I have defined sequentially closed and that closedness is something else. They're right, but for $\mathbb{R}^n$ they coincide).

As for openness, it is very different. A set is open if wherever you are in the set, you can find some comfortable area all around you that is still in the set. In $\mathbb{R}$ this corresponds to saying that around every point $x$, there is some interval $(a,b)$ that is a subset of the set and such that $x\in (a,b)$. You can rephrase this with sequences: if you take infinitely many steps and if you end up in the set, then you must have been in the set after finitely many steps. More formally: if you have a sequence $(x_n)_n$ that converges to $x$ and such that $x$ is in the set, then there is some $N$ such that $x_n$ is in the set for all $n\geq N$.

As you can see, closed and open are notions that are not at all extremes of each other, or inverses of each other. You can easily see $\mathbb{R}$ is open using these two explanations.

 

[The Bill]

As for openness, it is very different. A set is open if wherever you are in the set, you can find some comfortable area all around you that is still in the set.

I think using the words "all around you" is misleading here. I know you're trying to say that for every point in an open set one can find an open neighborhood set which contains the point and is in turn contained in the open set. However, since we're talking about topological spaces, an open neighborhood containing a point won't necessarily be intuitively "all around" that point.

Take the point 1 and the set [1,2) as a neighborhood of it in the Sorgenfrey line topology, for example. The set [1,2) isn't intuitively "all around" 1, but it is a neighborhood of 1 because the right half open intervals are defined as the basis of the Sorgenfrey line's topology, and therefore open.

Other than that quibble, I like your description of the false dichotomy of open and closed.

 

[micromass]

I think using the words "all around you" is misleading here. I know you're trying to say that for every point in an open set one can find an open neighborhood set which contains the point and is in turn contained in the open set. However, since we're talking about topological spaces, an open neighborhood containing a point won't necessarily be intuitively "all around" that point.

Take the point 1 and the set [1,2) as a neighborhood of it in the Sorgenfrey line topology, for example. The set [1,2) isn't intuitively "all around" 1, but it is a neighborhood of 1 because the right half open intervals are defined as the basis of the Sorgenfrey line's topology, and therefore open.

Other than that quibble, I like your description of the false dichotomy of open and closed.

You're aware the OP is in high school, right? Talking about exotic topologies won't really do her much good.

 

[The Bill]

You're aware the OP is in high school, right? Talking about exotic topologies won't really do her much good.

In post #3, ProfuselyQuarky mentioned topological spaces. That led me to believe we're talking about more than just the standard open ball topology on Euclidean space.

Also, more people than ProfuselyQuarky might read this thread in the future. I want to do what I can to preempt what confusion I can predict. I found this site by doing web searches related to questions I had.

 

[TeethWhitener]

Letting A⊆RA\subseteq \mathbb{R}, I know that a set A is open if every point of A is an interior point of A . A is closed, however, if and only if R∖A\mathbb{R}\setminus A.

This might have already been cleared up for you, but an example never hurts (right?). It turns out that in $\mathbb{R}$, the only clopen sets are $\emptyset$ and $\mathbb{R}$ itself. (This actually leads to another topological property known as connectedness.) However, as an example we can consider the set $(0,1)\cup(2,3)$ with the standard topology as a topological space. In this space, the complement of the open set $(0,1)$ is $(2,3)$, which is itself also an open set. But since it's the complement of an open set, that means it's closed, so it's both closed and open: clopen.

 

[ProfuselyQuarky]

There are many types of closedness in mathematics. But they all share the same characteristic. A set is closed if whatever you "do", you cannot go out of the set. Much like a prison. In topology, this translates to: if you are in the set and if you take infinitely many steps in the set, your position at infinity will still be in the set. More formally, if you have a sequence (xn)n(x_n)_n that converges to xx, then if xnx_n is in the set for each nn, then so is xx. (purists will no doubt tell me that I have defined sequentially closed and that closedness is something else. They're right, but for Rn\mathbb{R}^n they coincide).

As for openness, it is very different. A set is open if wherever you are in the set, you can find some comfortable area all around you that is still in the set. In R\mathbb{R} this corresponds to saying that around every point xx, there is some interval (a,b)(a,b) that is a subset of the set and such that x∈(a,b)x\in (a,b). You can rephrase this with sequences: if you take infinitely many steps and if you end up in the set, then you must have been in the set after finitely many steps. More formally: if you have a sequence (xn)n(x_n)_n that converges to xx and such that xx is in the set, then there is some NN such that xnx_n is in the set for all n≥Nn\geq N.

Thank yoouuuuu. This clears up a lot. I can really see how a set can be both open and closed now Regarding openness, "uncomfortable" area be where $x\notin(a,b)$, right?

This might have already been cleared up for you, but an example never hurts (right?). It turns out that in $\mathbb{R}$, the only clopen sets are $\emptyset$ and $\mathbb{R}$ itself. (This actually leads to another topological property known as connectedness.) However, as an example we can consider the set $(0,1)\cup(2,3)$ with the standard topology as a topological space. In this space, the complement of the open set $(0,1)$ is $(2,3)$, which is itself also an open set. But since it's the complement of an open set, that means it's closed, so it's both closed and open: clopen.

Thank you, too :) Looking at an example was helpful. I just looked to see what connectedness was and I've some questions about that, as well.

In post #3, ProfuselyQuarky mentioned topological spaces. That led me to believe we're talking about more than just the standard open ball topology on Euclidean space.

Apologies if quoting that part of the wiki article made it sound like I actually know about this topic. I do not

By the way (as a side question), the terms open and closed are really unfortunate (as micromass pointed out) and confusing. If you all could rename them, what would you choose? Obviously it has to stay "open" and "closed", but I like to come up with my own names/descriptions to scratch on the margins of my own notes just to help sometimes with tricky definitions and terminology.

 

[ProfuselyQuarky]

In a definition "if" is often used instead of more precisely "iff". In this case, a set is closed iff its complement is open; which means a set is open iff its complement is closed.

And thanks for clearing this up, too, PeroK!

 

[micromass]

Thank yoouuuuu. This clears up a lot. I can really see how a set can be both open and closed now Regarding openness, "uncomfortable" area be where $x\notin(a,b)$, right?

Depends. For example $(0,+\infty)$ is open. A comfortable area for $1$ would be $(0,2)$. Everything to the left of $0$ woldn't be in the set and wouldn't be comfortable.

Thank you, too :) Looking at an example was helpful. I just looked to see what connectedness was and I've some questions about that, as well.

By the way (as a side question), the terms open and closed are really unfortunate (as micromass pointed out) and confusing. If you all could rename them, what would you choose? Obviously it has to stay "open" and "closed", but I like to come up with my own names/descriptions to scratch on the margins of my own notes just to help sometimes with tricky definitions and terminology.

I like the name closed. Running in a sequence and remaining in the set really sounds like an awful closed prison. For open, perhaps something like "fat set" or "big set". That's something Carothers proposed. Of course, when you go to exotic spaces, the intuition breaks down, but in $\mathbb{R}^n$ surely, all open sets have this sense of "fatness". Maybe comfortable set or rich set? Doesn't do much good though, the name "open" is ingrained pretty deeply in mathematics.

 

[ProfuselyQuarky]

Everything to the left of $0$ woldn't be in the set and wouldn't be comfortable.

What about everything to the right of $2$? Same thing?

I like the name "fat set" and since I won't be delving deeper into exotic spaces any time soon, thinking of open sets as fat would be nice.

 

[micromass]

What about everything to the right of $2$? Same thing?

No, because they are in the set. So the set $(0,6)$ is "comfortable".

 

[ProfuselyQuarky]

No, because they are in the set. So the set $(0,6)$ is "comfortable".

It's just that you said that a comfortable area for $1$ would be the interval $(0,2)$ since $1\in(0,2)$. How is everything to the right of $2$ within the set?

 

[micromass]

It's just that you said that a comfortable area for $1$ would be the interval $(0,2)$ since $1\in(0,2)$. How is everything to the right of $2$ within the set?

We're talking about the set $(0,\infty)$ being open. This means that for every element in the set there must be a comfortable area (= an open interval). $(0,6)$ is an open interval inside the set $(0,\infty)$ so it's a comfortable area.

 

[ProfuselyQuarky]

We're talking about the set $(0,\infty)$ being open. This means that for every element in the set there must be a comfortable area (= an open interval). $(0,6)$ is an open interval inside the set $(0,\infty)$ so it's a comfortable area.

Oooh, that's what you meant. Thanks, micromass.