What is the topology of $E^*(E)$ where $E$ is a ring spectrum?

Question

In Adams' lectures on generalised cohomology Page 51, it states $E^*(E)$ is a topology ring. I do not know the topology on it. the reference that Adams offered there is by Novikov in Russian. Can anyone help to explain it?

Thanks!

Answer 1

Suppose we have spectra $X$ and $Y$. For any finite spectrum $W$, and any maps $X\xleftarrow{i}W\xrightarrow{f}Y$, we define $$ N(i,f) = \{g\colon X\to Y \;|\; gi=f\} \subseteq [X,Y]. $$ Sets of this type form a basis for a topology on the set $[X,Y]$. If $X$ is a CW spectrum (as it always is in Adams's book) then you can restrict attention to the case where $W$ is a finite subspectrum of $X$ and $i$ is the inclusion; this will give the same topology. Note that when $X$ itself is finite the topology is discrete, so the definition is only really interesting in the infinite case.