by Vadim Kosoy 19 days ago | link | parent Very nice. I wonder whether this fixed point theorem also implies the various generalization of Kakutani’s fixed point theorem in the literature, such as Lassonde’s theorem about compositions of Kakutani functions. It sounds like it should because the composition of hypercontinuous functions is hypercontinuous, but I don’t see the formal argument immediately since if we have $$x \in *X,\ y \in *Y$$ with standard parts $$x_\omega,\ y_\omega$$ s.t. $$f(x)=y$$, and and $$y' \in *Y,\ z \in *Z$$ with standard parts $$y'_\omega=y_\omega,\ z_\omega$$ s.t. $$g(y')=z$$ then it’s not clear why there should be $$x'\in X,\ z'\in Z$$ s.t. with standard parts $$x'_\omega=x_\omega,\ z'_\omega=z_\omega$$ s.t. $$g(f(x'))=z'$$.

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