Suppose a set $X$ has the property that for all $a,b \in X$, we have that $a=b$. Such a set is either empty, or it's a singleton set. In other words, it has one or fewer elements. What do we call this?
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I've occasionally used "subsingleton". I think this is OK in the context of classical logic, which is what you're using since you say that such a set must be a singleton or empty. In the context of constructive logic, "subsingleton" doesn't look so good, since it suggests that the set is a subset of some singleton, which need not be the case. Unfortunately, I don't know a better term that avoids constructively unwanted suggestions. 


I would explain the concept and then name it. Since the idea is to capture the uniqueness of any potential member of the set X, I would suggest describing such X as "singular". Be sure not to use "singular" for anything else in that context. Gerhard "Caution: Alternate Words At Work" Paseman, 2012.09.20 


In category theory, subobjects of the terminal object are sometimes called subterminal. An equivalent definition: an object $A$ is subterminal if and only if for all objects $B$, there is at most one map $B \to A$. (Note that in the case where the category is $\mathbf{Set}$ and $B = 1$, this says exactly that $A$ has at most one element.) The sets you mention are the subterminal sets. (But still, if I was writing a paper in which I only mentioned such sets a handful of times, I'd probably just say "sets with at most one element".) 


The language of homotopy type theory suggests names for a set with the "any two objects are equal" property. You can call it a $(1)$groupoid or a $(1)$type or a $h$proposition. It may sound like the names involving $1$ suffer from the same weaknesses as Asaf's suggestion set of size at most one, namely that it depends on numbers. But in fact there is an inductive definition of $n$type (starting with $2$), and it is easy to unpack it to get this definition. Unfortunately, to many audiences these names are currently likely to cause more confusion than joy. Still, maybe someone should try to popularise them outside their current domains. 

