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.