2
$\begingroup$

Is there an example of a non-compact and connected space $X$ such that constant functions are the only continuous functions from $X$ to $\mathbb{R}$ with the usual topology?

If the answer is "yes", I wonder under which conditions a non-compact and connected space possesses a non-constant continuous real-valued function.

Here are the related questions: Existence of non-constant continuous functions and Topological space $X$ which the set of non-constant real-valued continuous function on $X$ is empty.

Thank you.

$\endgroup$
3
$\begingroup$

Here is a easy example: An uncountable $X$ equipped with the cocountable topology is connected noncompact and the only continuous real-valued functions are constants. The proof is almost the same as the cofinite case.

Indeed, to admit a nonconstant $\mathbb{R}$-valued function means there is a nontrivial Hausdorff quotient $X\to f(X)$, and of course you can impose compactness there too since you can cut off $f$.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ You can also see that if $f:X\to \Bbb R$ is continuous and $f(x) \ne f(y)$ then $f^{-1}(-\infty, (f(x)+f(y))/2)$ and $f^{-1}((f(x)+f(y)/2,\;\infty)$ are disjoint non-empty subsets of $X$, which is not possible ...........+1 $\endgroup$ – DanielWainfleet Sep 18 '18 at 19:24

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.