How can I prove that the function space $\mathcal{C}[0,1]$ of all continuous real valued functions on $[0,1]$ with the sup metric is connected?
I think the sup metric is as follows:
If $f, g $ are in $\mathcal C [0,1]$, then $$d(f,g)= \sup_{x\in [a,b]} |f(x)-g(x)|$$
To show that it's connected, I think we can better prove that it's path connected, which implies connectedness.
Hint: To show that the space is path connected, we need some continuous $\gamma : [0,1] \rightarrow \mathcal{C}[0,1]$ such that $\gamma(0)=f(x), \gamma(1) = g(x)$. (Since $\forall t \in [0,1]$, $\gamma(t)$ is a continuous function, denote $h_t(x)$ to be the continuous function given by $\gamma(t)$.)
Define $\gamma(t) =h_t(x) = f(x) + t(g(x)-f(x)).$ It is easily shown that this is continuous (in fact, uniformly continuous).
The idea of "smoothly transforming" one function into another is much used in algebraic topology(amongst other fields), and the name of the path joining two functions is called a "homotopy" betwen them.
