2
$\begingroup$

Is it true that every Baire function $f:\mathbb{R}\to \mathbb{N}$ must be constant?

$f:X \to Y$ is a Baire function for $X,Y$ metrizable spaces if $f$ is a member of $F(X,Y)$, where $F$ is the smallest class of function from $X$ to $Y$ containing continuous functions and being closed under pointwise limits.

$\endgroup$
  • 1
    $\begingroup$ How do you define a Baire function from $\Bbb R$ to $\Bbb N$? Have you tried writing down the definitions in full, and see if the answer is positive? $\endgroup$ – Asaf Karagila Aug 16 '13 at 4:18
3
$\begingroup$

HINT:

Recall that the continuous image of a connected space is connected. Therefore every continuous function from $\Bbb R$ to $\Bbb N$ is constant. It suffices to show, if so, that pointwise limit of constant functions is constant.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.