The space $C[0,1]$ is path connected.

Question

I have a proof of the statement. Can you please let me know where did I go wrong?

The space $C[0,1]$ is path connected. Proof: Let $f,g\in C[0,1].$ Define

$T:[0,1] \to C[0,1]$ s.t $T(t) = f(x) +t(f(x) - g(x))$ so $T(t) \in C[0,1]$

NOW I will show $T(t)$ is continuous.when $t_n \to t$ , $T(t_n) \to T(t)$ as $||T(t_n) - T(t)|| \leq |t_n - t|.||(f(x) - g(x) )||$.(It is possible because $[0,1]$ is haudroff space).

That's how $C[0,1]$ is path connected.

Please mention where I went wrong.

Answer 1

There is nothing wrong with it. The definition of $T$ is redundant (if you define $T(t)=f(x)+t\bigl(g(x)-f(x)\bigr)$, there is no need to say that $T(0)=f(x)$ and that $T(1)=g(x)$), but that is not a logical error.