I'm working on a problem which states the following:
Given the set $A=[-2,-1] \cup [1,2]$, is $C[A]$ (continuous functions from $A$ to $\mathbb{R}$) a connected space? The metric being used in the question is $d_\infty (f,g) = \sup\{ |f-g| | x\in X \}$.
My intuiting tells me that it is a connected space, but I can't seem to prove it. I was thinking about using path connectedness. Am I correct in saying that as the functions in $C[A]$ are continuous, they reach a maximum and minimum value and therefore are bounded sequences. I can then use the argument in this question and conclude that the space is path connected and therefore also connected.
Is this correct? Is there a way to solve this question without using path connectedness? If so could someone show that solution?
