I have a question relating to an answer on MathOverflow.net. The cited answer says:
Let $X$ be a topological space for which [the Euler characteristic] $\chi(X)$ is defined and behaves in the expected way for unions, Cartesian products, and quotients by a finite free action. ... [Then] $\chi(X^{(2)}) = \frac{\chi(X \times X) - \chi(\operatorname{Diag}(X))}{2} + \chi(X) = \frac{\chi(X)^2 + \chi(X)}{2}$ [where $X^{(2)}$ denotes the symmetric square of $X$].
Question: Does anyone know a reference for this result, or, failing that, a short proof? For the application that I have in mind I need the result for algebraic varieties over an algebraically closed field (whose characteristic may be positive), but a more general result would be nice to see.