# Prove that C(X) is path-connected

• May 20th 2009, 06:45 PM
h2osprey
Prove that C(X) is path-connected
Given a metric space (X, dx), let C(X) denote the set of continuous functions from X to $\displaystyle R$. For two functions $\displaystyle f, g$ in C(X), define

d($\displaystyle f,g$) = sup |$\displaystyle f(x) - g(x)$| for x in X.

Prove that C(X) is path-connected.

Thanks!
• May 20th 2009, 08:34 PM
TheEmptySet
Quote:

Originally Posted by h2osprey
Given a metric space (X, dx), let C(X) denote the set of continuous functions from X to $\displaystyle R$. For two functions $\displaystyle f, g$ in C(X), define

d($\displaystyle f,g$) = sup |$\displaystyle f(x) - g(x)$| for x in X.

Prove that C(X) is path-connected.

Thanks!

We need to show two things.

First we need a path between and two points.

Remember a path is $\displaystyle f:[0,1] \to X$

$\displaystyle p(t)=(1-t)\cdot f +t\cdot g$

note that $\displaystyle p(0)=f$ and $\displaystyle p(1)=g$.

Now we need to show that the path stays in C(X)

If you know the right theorems i.e products and sums of continous functions are continous then you are done, but my guess since the gave you the metric is that you need to show $\displaystyle p(t)$ is continous.

Either use a seqence or delta epsilon proof. for cont.

I hope this helps.