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.

I would have said that the suspension was $X \times [-1, 1]$ modulo the relation that 
$$
(x, a) \sim (x', a') 
$$
if and only iff 


*

*$a = a' = 1$ or 

*$a = a' = -1$, or

*$a = a'$ and $x = x'$. 


Wikipedia seems to agree with me. It looks as if your author was a little glib, and failed to mention that the "bottom" and "top" sets of equivalent points were not supposed to be made equivalent to each other. 
