Which of the statements is wrong:


1. a generalized cohomology theory (on well behaved topological spaces) is determined by its values on a point
2. reduced complex $K$-theory $\tilde K$ and reduced real $K$-theory $\widetilde{KO}$ are generalized cohomology theories (on well behaved topological spaces)
3. $\tilde K(\*)= \widetilde{KO} (\*)=0$


But certainly $\tilde K\neq \widetilde{KO}$.



K-theory as a generalized cohomology theory
1 is doubly wrong. First, you need to distinguished generalized cohomology theories and *reduced* generalized cohomology theories. If you want to work with the latter, you should replace "a point" in 1 by "$S^0$", and then the corrected version of 3 no longer holds. But even this new version 1' is false; a generalized cohomology theory is not determined by its coefficients, unless they are concentrated in a single degree (example: complex K-theory vs. integer cohomology made even periodic).


