Clopen set question (topology)?

Question

A clopen set is one which is both open and closed.

Does this simply mean that it is open at one 'end' and closed at the other?

And as a follow-up, can anyone recommend a source (book or print/online journal) which provides a definition of 'clopen'. Currently I only have Wikipedia, which is not scholarly enough for my purposes.

Or does it mean that it is both closed and open (simultaneously) at the internal boundary?

S**t! Sorry for mucking up the paragraphs above!

Thanks for your great anwer Pascal.

The definition is for my PhD thesis (that's why I'm not so keen to cite Wikipedia, even though its maths articles are pretty good).

Answer 1

No, it means exactly what it says - the set is both open and closed. Recall that the definition of a closed set is one whose complement is an open set (and thus, closed is not the opposite of open). A half-open interval, such as [0, 1), is neither open (there is no open neighborhood of 0 in the set), nor closed (the complement of [0, 1) is not open, because there is no open neighborhood of 1 in the complement of [0, 1)). Therefore it is certainly not clopen.

In any topological space whatsoever, both ∅ and the entire set are clopen (to see this, note that one of the axioms that a topology on X is required to satisfy is that both ∅ and X are open sets, and since the complement of ∅ is X and the complement of X is ∅, their complements must also be open as well, so they are also closed sets and thus clopen). Now, on ℝ with its usual topology, these are also the only clopen sets (see proof at end), which is one of the reasons why students tend to think that closed and open are opposing concepts. This is actually a peculiarity of connected spaces, and not true of topological spaces in general. For instance, on ℚ, the interval (-∞, √2) is clopen (its complement is (√2, ∞) rather than [√2, ∞), courtesy of the irrationality of √2).

As for whether this means that it is both closed and open at the boundary -- in a sense, yes. The boundary of a set is defined as the closure of S minus the interior of S. The closure of a closed set is itself and the interior of an open set is itself, so the boundary of any clopen set S will simply be S\S, which is ∅, and ∅ is certainly both open and closed. Note however that ∂S is clopen does not imply that S is clopen. However, the implication ∂S = ∅ → S is clopen DOES hold, so if you want to think of clopenness in terms of the boundary points, a clopen set is simply one whose boundary is the empty set.

By the way, what are your purposes? If you're only trying to learn, then wikipedia is more than scholarly enough (it is almost always accurate). If you're trying to write a paper, then it's true that you (probably) won't be allowed to cite wikipedia, but then you shouldn't need to provide a citation for the definition of a clopen set, any more than you should need to cite 1+1=2 -- the definition of a clopen set is one of those things which are considered to be common knowledge in topology.

Proof that the only clopen sets on ℝ are ∅ and ℝ: let S⊂R such that S≠∅ and S≠ℝ. Then ∃x∈S and ∃y∉S. Assuming WLOG that x0 : |z-c|<δ → x∈S. Also, c≠y since c∈S and y∉S, thus y-c > 0. So then z=c+min(δ, y-c) > c, but |z-c|<δ so z∈S, so c fails to be an upper bound of {z∈[x, y]: z∈S} - a contradiction. Therefore, if c∈S, S cannot be open. Conversely, suppose c∉S. If ℝ\S is open, then ∃δ>0 : |z-c| > δ → z∉S. We know δ ≤ c-x, since x∈S. But then c-δ is a lower upper bound for {z∈[x, y]: z∈S}, a contradiction. So if c∉S, ℝ\S cannot be open. In either case, either S or its complement is not open, so S is not clopen. Q.E.D.)