1. ## Simple Question

if $\displaystyle X$ is a topological space, is $\displaystyle X \times [0,1]$ a topological space, where $\displaystyle [0,1]$ is the unit interval?

Would $\displaystyle X \times [0,\frac{1}{2}] \subseteq X \times [0,1]$ be closed or open or both?

I am interested because im trying to prove that homotopy is an equivalence relation (Transitivity)

Thanks for any help

2. Originally Posted by slevvio
if $\displaystyle X$ is a topological space, is $\displaystyle X \times [0,1]$ a topological space, where $\displaystyle [0,1]$ is the unit interval?

Would $\displaystyle X \times [0,\frac{1}{2}] \subseteq X \times [0,1]$ be closed or open or both?

I am interested because im trying to prove that homotopy is an equivalence relation (Transitivity)

Thanks for any help
You mean $\displaystyle X\times [0,1]$...wrt what topology?!

Tonio

3. If you mean that $\displaystyle [0, 1]$ has the "usual" metric topology on the real numbers, restricted to $\displaystyle [0, 1]$, and that $\displaystyle X\times [0, 1]$ is to be given the usual "product topology", then, yes, of course, it is a topological space. If A and B are are any two topological spaces then $\displaystyle A\times B$ "with the product topology" is, by definition, a topological space. Since $\displaystyle [0, 1/2]$ is not an open set in the usual topology on $\displaystyle [0, 1]$, no, $\displaystyle X\times [0, 1]$ is not an open set. Since $\displaystyle [0, 1/2]$ is a closed set in the usual topology on $\displaystyle [0, 1]$, yes, $\displaystyle X\times [0, 1/2]$ is a closed set.

4. I think you mean $\displaystyle X \times [0,\frac{1}{2}]$ is not an open set.

But thanks for your post this clears things up for me!!