I'm aware that R is not the only set that satisfies the least upper bound property, the p-adics do also. Does the intermediate value theorem also hold in the p-adics then?What are the spaces where the intermediate value theorem hold?

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I&#039;m aware that       R   is not the only set that satisfies the least upper bound property, the   p-adics do also. Does the intermediate value theorem also hold in the   p-adics then?What are the spaces where the intermediate value theorem hold?

Julien Carrillo · Accepted Answer

The intermediate value theorem can be generalized for topological spaces. For me, this makes sense, since the hypotheses for the IVT are in essence that given a function   f  :  X  →  Y, where   X  ,  Y are topological spaces and1.   X is connected2.   f is continuous3. There is some ordering on   (  Y  ,  &amp;lt;  ) (equipped with the order topology)then the intermediate value theorem holds.We need hypothesis 1 and 2 to make sure that (heuristically at least) there are no &quot;jumps&quot; in the values our function takes. For example,   f  :  (  −  ∞  ,  0  )  ∪  (  0  ,  ∞  )  →      R   given by   f  (  x  )  =  1      /    x is continuous, but its domain is disconnected, so we cannot guarantee that the function takes on the value 0. On the other hand take   f to be any step function on a connected domain, and note that it fails to satisfy the intermediate value theorem.Hypothesis 3 is there simply to make sense of the theorem anyway. Our IVT for       R   can be recovered by noting that the connected components of the real line are exactly the rays and intervals.If I&#039;m not mistaken, the   p-adic rationals are totally disconnected, so it seems unlikely that there would exist a generalization to this setting.

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