TITLE: K-theory as a generalized cohomology theory
QUESTION [11 upvotes]: Which of the statements is wrong:

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

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

REPLY [17 votes]: 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).