In Lee's Introduction to topological manifolds on page 340 he writes that an element of $C_p(X)$ can be written as a formal linear combination of singular $p$-simplices. Similarly, on Wikipedia's entry on singular homology the chain groups are described as formal linear combinations of singular simplices with integer coefficients.
But to me it seems that $C_p(X)$ are just linear combinations with integer coefficients. What am I missing? What does formal mean here?
Before you form a free abelian group on a set S, there is no addition operation with which to form combinations. So to build it, you have to describe elements of the space formally as "just looking like this sum."
Then you are free to define an addition operation (and a scaling operation, if we're talking about a vector space) and we have a bona fide algebraic object in which the formerly formally defined sums are now "actual" sums.
So you see, before you create the group, there is no way to initially describe it using "actual" sums. That's the temporary reason we need to say "formal."
