Here is the quesion: Let be $X$ a topological space with trivial topology $\mathcal{T} = \{\emptyset,X\}$. Prove $X$ is contractil (any constant map $f(x) = x_0$ is homotopical to the identity map $Id$).
WAY #1: Define $F : X \times [0,1] \to X$ as $F(x,0) := x$ and $F(x,t) := x_0$ for $t \in (0,1]$. $F$ is continous (cause $\mathcal{T}$ is trivial) and satisfies $F(x_0,t) = x_0$ (is it necessary this one?), $F(x,0) = Id$ and $F(x,1) = x_0,$ so it is an homotopy.
WAY #2: We define $F : X \times [0,1] \to X$ as $F(x,t) := x_0t + (1-t)x$. Again, $F$ is continous (cause the topology of $X$) and satisfies $F(x,0) = Id$, $F(x,1) = x_0,$ so it is an homotopy.
What way is right?
Thank you all.
Way #1 is right and way #2 is not. The (is it necessary this one?) was indeed not necessary. Way #2 is not right, because $X$ need not be a vector space, so $x_0t+(1-t)x$ need not be well-defined.