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It is usually argued (and also joked about) that classifying sets into open and closed is a bit paradoxical, since sets can be open and closed at the same time, or neither. This can be analyzed very clearly by noting that closed is an antonym of open: it means exactly not open (and vice versa). Then, by saying that a set is clopen, or open and closed, we are in some way claiming that the set is open and not open, and this is a basic contradiction from a logical point of view.

While I don't dare to think that I can make an impact on the use of these terms by changing the way I myself call some sets, it would be much easier for me to at least think of them with different names. Particularly, I've been wondering about the correctness of saying co-open instead of closed.

In the first place, it makes sense because it's more natural to see the link between declaring $A$ is co-open and the complement of $A$ is open. Also, since complementation is an involution, in some natural way we can say that a co-co-open set, which we may understand as a set whose complement has an open complement, is nothing but an open set; this, I believe, is desirable behavior for the use of the co- prefix.

My final worry about the correctness of using co-open is that the co- prefix is widely used in category theory, and is formalized by the notion of duality (to name but a few examples: initial and coinitial, or terminal and coterminal, product and coproduct, limit and colimit, cone and cocone). So I've been wondering whether there should be some categorical formulation of a topology so that saying $A$ is co-open in $\mathcal{C}$ was equivalent to saying $A$ is open in $\mathcal{C}^{op}$, where $\mathcal{C}$ was the categorical formulation of a topology in which $A$ is a closed set. This, in order to justify the use of co- with duality.

However, I am aware of the existence of other terms which start with the prefix co- but, I believe, don't have a possible categorical formulation or which were named before they were redefined in terms of categorical notions, like cologarithm, cosine, cosecant, cotangent, cotree.

My question would then be: aside from the issue that every mathematician writes closed, could co-open be considered correct as a terminological alternative, based on the points I exposed, and on others I might possibly have missed? At the end, if co-open is correct, this will be only useful for my own pedagogical reasons.

Thanks.

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reminds me of this (slightly) related youtube video: youtube.com/watch?v=SyD4p8_y8Kw –  Aaron Mazel-Gee Apr 4 '11 at 7:16
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Why not “closed” and “coclosed”? “Closed” and “closure” en.wikipedia.org/wiki/Closure_operator are derived from the same word. “Open” and “interior” from different. IMHO it is better to keep words with 1 root. –  beroal Apr 5 '11 at 16:13
    
The link between open and closed sets is related to en.wikipedia.org/wiki/Duality_%28order_theory%29 and dual logical connectives. –  beroal Apr 5 '11 at 16:14
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This can be analyzed very clearly by noting that closed is an antonym of open: it means exactly not open (and vice versa)

This is confusing. Closed doesn't 'mean not open, it means the complement of an open set. So a set that is both open and closed is one which is open and for which its complement is open.

I think the terminology is good because we want a closed set to be one for which every convergent sequence in that set converges to a limit that is also in that set. We want it to be closed under convergence and this can only be the case if its complement is open.

So, closed is a valuable word in this context. If anything, I would replace open. I'm just beginning with this stuff though so there might be something that I'm missing.

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I meant that the literal meaning of closed is not open, not the specific mathematical definition. I see your point with the usefulness of closed. Basically, a set that contains its boundary. It might be the case that in certain circumstances it could be better to think of a set as co-open as I defined it, and in others to consider it closed, referring to different aspects of the concept according to which are more important in a given setting, and making the obvious identification. This is, after all, only about how I find convenient to think of these concepts. –  Abel Apr 3 '11 at 23:37
    
@Abel, right. They're two different things and I don't see what one has to do with the other. It seems like comparing the two is just a cause for confusion. Like I said, I'm just a nobody though. –  knucklebumpler Apr 3 '11 at 23:39
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I suspect that Abel was talking about the regular dictionary definition which, in my dictionary, is "closed. adjective. not open" –  George Lowther Apr 3 '11 at 23:40
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I guess I'm saying that closed in the mathematical sense and closed in the English sense are two entirely different words that only happen to share a spelling and pronunciation. So the literal meaning of closed is not not open but the complement of an open set. –  knucklebumpler Apr 3 '11 at 23:44
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@knucklebumper: I agree, but I'm arguing that some confusion stems from that particular choice of words. –  Abel Apr 4 '11 at 0:11
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