Definition of cone over $X$ The cone over $X$, as we know, is defined as $CX=X\times I/\{x\}\times I$ where the geometric picture is clear to me. But I saw another definition where $CX$ is defined as $X\times I/\{x\}\times I\cup X \times \{1\}$ which is not visually very obvious to me. Can you please explain why these two definitions are equivalent?
 A: The cone over a space $X$ is defined as $CX = X\times I ~/~ X \times \{1\}$.
The reduced cone over a pointed space $(X, x_0)$ is defined as $\overline{CX} = X\times I ~/~ (X \times \{1\} \cup \{x_0\}\times I)$. It is also a pointed space with the obvious choice of basepoint.
These two definitions are not equivalent. However if $(X, x_0)$ is a well-pointed space, then so is $\overline{CX}$, and $CX$ is homotopy equivalent to $\overline{CX}$.
Here's how to visualize these constructions:
Taking the product of a space and an interval yields a cylinder. Contract one face of that cylinder to a point, and you get the cone of your space.
Taking the product of a pointed space and an interval yields a cylinder, the induced fiber of the basepoint is a line along the side of the cylinder. Now contract one face of that cylinder to a point, and also contract that line to the same point. You'll get a weird cone-ish thing that is pinched together over the basepoint. That is the reduced cone of your space.
