1
$\begingroup$

What is the closure of the set of continuous bounded real-valued functions on $\mathbb{R}^d$ under pointwise convergence? How might one go about finding this closure? Also, are there common names for spaces of functions equipped with different notions of convergence (topologies)?

Does anyone have a comprehensive resource listing results like the closures of common function spaces with certain topologies? Thanks.

$\endgroup$
3
$\begingroup$

Such functions (pointwise sequential limits of continuous funvtions) are called Baire functions of the first class. (the second class is are the pointwise limit of those etc.), see Wikipedia or this survey paper. They are quite well-studied and classical objects in topology.

It’s well-known that the continuous functions are dense in all functions on a space $X$, provided that $X$ has enough real continuous functions, e.g when $X$ is Tychonoff or even just functionally Hausdorff. So the fact I said sequential limits is quite important.

$\endgroup$
  • 1
    $\begingroup$ You mean pointwise limits of SEQUENCES of continuous functions. The space of continuous functions is dense in the space of all real-valued functions in the topology of pointwiese convergence. $\endgroup$ – Jochen Mar 17 '19 at 17:18
  • $\begingroup$ @Jochen true, in Tychonoff spaces. $\endgroup$ – Henno Brandsma Mar 17 '19 at 17:20

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.