Let $X$ be a topological space, and $C(X)= (X \times [0 ,1])/(X \times {1} )$,
define $f\colon X \to C(X)$ as $f(x)=[x,t]$ for some fixed $t$ s.t $\ 0\leq t <1$.
I have to show this is a continuous homemorphism onto its image. Injectivity is easy, and continuity comes from the quotient topology on cone.
I have problem showing that this is an open map. I start with an open set $O \subset X$ then $f(O) = \lbrace f(x) \vert x \in O \rbrace = \lbrace [x,t] \vert x\in O \rbrace =\lbrace (x,t) \vert x\in O \rbrace= O \times \lbrace t \rbrace$
How do I show this set is open in the cone?