In Segal's paper on $\Gamma$-spaces, he gives a functor $Spectra \rightarrow \Gamma-Spaces$ defined by taking a functor $E$ and sending it to the $\Gamma$-space $AE$ with $AE(n) = Mor(S \times \cdots \times S, E)$, where $S$ is the sphere spectrum. Now, since this is supposed to define a $\Gamma$-space, in particular the sets $Mor(S \times \cdots \times S, E)$ should be topological spaces... but they don't seem to come with any obvious topology, at least not obvious to me.


On the other hand, it seems like there should be some sort of spectrum that acts like $Mor(S \times \cdots \times S, E)$; could he mean, possibly, the 0th space of this spectrum?


EDIT: The reference is 


Segal, Graeme Categories and cohomology theories. Topology 13 (1974), 293--312



What is the topology on hom-sets of spectra?
If $X = (X\_n)$ and $Y = (Y\_n)$ are spectra, one can define a morphism just to be a collection of maps $x\_n \to Y\_n$ commuting with the suspensions. Thus the set of morphisms between $X$ and $Y$ is a subset of $\prod\_n Map(X\_n,Y\_n)$ - and we give it just the subspace topology. 


An alternative way is the simplicial set approach: we define an $n$-simplex of the mapping space between $X$ and $Y$ to be a map $X\wedge \Delta[n]^+ \to Y$. If we want a topological space back, one can geometrically realize. 


If you want mapping spectra, it is perhaps more reasonable to go to symmetric spectra. You can find an exposition of mapping spectra (and also of mapping spaces) in Stefan Schwede's book project on symmetric spectra: <http://www.math.uni-bonn.de/people/schwede/SymSpec.pdf> 2.24 & 2.25.


If you're interested in the relationship between Gamma-spaces and spectra from a homotopical view, you might also be interested in the Bousfield-Friedlander paper: <http://club.pdmi.ras.ru/~topology/books/bousfield-friedlander.pdf>


