if is a topological space, is a topological space, where is the unit interval?
Would 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
If you mean that has the "usual" metric topology on the real numbers, restricted to , and that 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 "with the product topology" is, by definition, a topological space. Since is not an open set in the usual topology on , no, is not an open set. Since is a closed set in the usual topology on , yes, is a closed set.