I'm reading the paper ['The big fundamental group, big Hawaiian earrings and the big free groups'](http://www.sciencedirect.com/science/article/pii/S0166864199001042). The authors state that the class of homotopy equivalences of loops in the space he dubs as the big Hawaiian earrings is not a set. I'm not really certain if what he is saying is true. 


I think my question amounts to asking is the class of functions from a set to another set not always a set?


The construction he gives of the big Hawaiian earrings (for ordinal $\omega$) is as the one point compactification of the set of disjoint open intervals with cardinality $\omega$. He then maintains that if $\omega$ has cardinality greater than $\aleph\_0$ the resulting class of equivalence classes of homotopic loops is not a group since it's not a set. Is this true? I thought the Hom class of maps between all sets was a set. I'd appreciate any insight, thanks.



When is the class of functions between sets a set?
To answer the question, putting together points made in the comments...


There is always a set of functions between two sets, unless you are using [predicative foundations](http://ncatlab.org/nlab/show/predicative+mathematics), where there may be no function sets, or the set theory [NF(U)](http://en.wikipedia.org/wiki/New_Foundations) which is classical (even [material](http://ncatlab.org/nlab/show/material+set+theory)!) but famously has no function sets.


However, this is not what the authors of the article are considering, rather they are considering (in ordinary classical foundations) a disjoint union 
$$
\coprod\_{L\in BigIntervals} Top\_\ast(L,X)
$$
where each $Top\_\ast(L,X)$ is a set. For Hausdorff $X$, the authors state 



> 
> "We will be saved from set-theoretic difﬁculties only by the remarkable fact that all maps deﬁned on big intervals into a ﬁxed space $X$ factor through maps deﬁned on relatively small big intervals."
> 
> 
> 


Thus they are considering the quotient of the above disjoint union by a relation 'factors through'. That is, for a fixed $X$, there is a *set* $\{L\_\alpha\}\_{\alpha\in I}$ of big intervals such that every $f\colon L\to X$ for an arbitrary big interval $L$ factors through some $f\_\alpha\colon L\_\alpha \to X$, and then $f \sim f\_\alpha$. That this isn't a problem is resolved by some variant of [Scott's trick](http://en.wikipedia.org/wiki/Scott%27s_trick), which allows for taking quotients of proper classes, or by arguing directly that the above disjoint union should be replaced by
$$
\coprod\_{L\_\alpha\in I} Top\_\ast(L\_\alpha,X).
$$


In the event that one really has a proper class, then the fact that there is a group structure (class function giving multiplication etc) on that class means that it is a group, just not a set-sized group. Some people (e.g. Conway) would call this a Group, to distinguish it; category theorists might call it a 'large group', if using the terminology of 'small' and 'large' to distinguish set-theoretic size issues.


