Consider the set $G=\{ f : [0,1] \rightarrow [0,1] : f$ is an increasing homeomorphism $\}$, with the compact-open topology. I am trying to prove that $G$ is contractible.
What I tried so far, given any $f \in G$, there is a homotopy between $f$ and the identity function $id$, explictely, this homotopy is given by $F: I \times I \rightarrow I$, $F(x,t) = tx + (1-t)f(x)$; visually I know that I am deforming $f$ into the line $y = x$ under the line connecting $(x,x) $ and $(x,f(x))$.
I would like to define the homotopy $G \times I \rightarrow G$ by $(f,t) \mapsto f_t = F( , t)$ to conclude that $G$ is contractible; however, I am not able to prove that $f_t \in G$, I know that they are homeomorphism.
Well, you just have to show $f_t$ is increasing. If $x<y$, then $f_t(x)=tx+(1-t)f(x)$ and $f_t(y)=ty+(1-t)f(y)$. Since $f$ is increasing, $f(x)<f(y)$, so $$f_t(x)=tx+(1-t)f(x)<ty+(1-t)f(y)=f_t(y),$$ and you're done.