unreduced suspension Is the definition $SX=\frac{(X\times [a,b])}{(X\times\{a\}\cup X\times \{b\})}$  of the unreduced suspension the standard defininition? If I consider $X=$ point, the suspension of $X$ is a circle. But I saw an other definition of the unreduced suspension such that the suspension of a point should be an interval. Regards.

In fact in my experience the unreduced suspension is commonly written in the way you did it, although it is wrong. What it means is: collapse one side to a point and also the other side. But NOT both to the same point. So this would be only right when quotiening is associative, i.e.
$$
SX = (X \times I/ X\times \{0\} )/ (X \times \{1\}) "=" X \times I / X\times 0 \cup X\times 1
$$
To answer your question clearly: no this definition is wrong (or at least not standard)!
