Franklin's Notes

Intermediate value property

A function $f:A\to B$ between linearly ordered sets is said to have the intermediate value property if $f(a)<b<f(a')$ implies that there exists $a''\in A$ such that $f(a'')=b$ (from which it follows that $a<a''<a'$).

If $A,B$ are linear orders, let us say that $A$ is sparse over $B$ if there exists no non-constant monotone function $f:A\to B$ with the intermediate value property. For instance, $\mathbb R$ is sparse over $\mathbb Q$ is sparse over $\mathbb N$ is sparse over $[1]$.


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