3

Earlier this year, thanks to various users on this site, I was able to answer a question dealing with the properties of $C_{p}([0,1])$ here: Properties of Cp(X).

$C_{p}([0,1])$ is the space of all continuous real-valued functions on $[0,1]$ with the topology of point-wise convergence.

I was looking at problems dealing with this space, and I came across these two:

Prove that the space $C_{p}([0,1])$ is Lindelöf.

Is $C_{p}([0,1])$ normal?

Can anyone help me out with this one? Thanks in advance!

Community
  • 1
Maria
  • 2,099
  • 3
    Can you find a Lindelöf space that maps continuously onto $C_p[0,1]$? (Hint: there is one in the thread you link to). Recall (or prove): if $X$ is Lindelöf and $Y$ is regular and $f\colon X \to Y$ is continuous then $Y$ is Lindelöf. – t.b. Sep 03 '12 at 01:32
  • If you show it is Lindelöf, then (every $C_{p}(X)$ being regular, even completely regular) it is also normal. So it's only the first that you need to concern yourself with. – Henno Brandsma Sep 03 '12 at 05:55
  • Hint: if $X$ has a countable network, so does $C_p(X)$. A network is like a base, but the members need not be open sets. A regular space with a countable network is Lindelöf... – Henno Brandsma Sep 03 '12 at 06:02
  • @t.b. I'm not seeing which space you are talking about from the thread I linked to. Can you give me any more hints or suggestions? – Maria Sep 03 '12 at 17:32
  • 2
    The space $C[0,1]$ with the $\sup$-norm is a separable Banach space, hence second countable and thus Lindelöf. The identity mapping $C[0,1] \to C_p[0,1]$ is continuous. – t.b. Sep 03 '12 at 17:47
  • @t.b. Thank you! You have helped me greatly! – Maria Sep 03 '12 at 19:12

1 Answers1

5

Here's how I would argue (summarizing the comments):

  1. The space $C[0,1]$ with the $\sup$-norm is a separable Banach space, hence it is second countable and thus Lindelöf. The identity mapping $C[0,1] \to C_p[0,1]$ is a continuous bijection, so $C_p[0,1]$ is Lindelöf (if your definition of Lindelöf includes regular, use the first sentence in 2. in addition).

  2. $C_p[0,1]$ is regular because it is $\mathbb{R}^{[0,1]}$ with the product topology. A regular Lindelöf space is paracompact and hence normal.

Alternatively for 1., you could take the route suggested by Henno in the comments and prove the stronger property that $C_p[0,1]$ has a countable network, hence it is Lindelöf.

t.b.
  • 78,116