I am a bit confused with the definition of a suspension. This the definition.
For a space $X$, denote $SX$ the suspension of $X$ in which this is the quotient space $$\frac{X \times I}{\sim}$$ where $\sim$ is the equivalence of relation of $X \times \{0\}$ and $X \times \{1\}$ collapsed to a point.
The typical example is to set $X = S^n$. For $n = 1$, this is a "cylinder". What I don't understand is that why when we collapsed the top and end point of the cylinder our quotient space immediately becomes a "double-cone"? For example let's say $X \times \{1\} \to \{x_1 \} \times \{1 \}$ and $X \times \{0\} \to \{x_2\} \times \{0\}$. I don't understand why suddenly points close to $0$ and $1$ "shrink". For example, at $X \times \{3/4\}$, the "cone' picture depicts $S^1$ with a smaller radius.
To clarify what the problem is when we "shrink", at $\{3/4\}$ $X$ is no longer $S^n$
If the above example is too diffuclt to explain, we can work with $X = I$, so that $(X \times I) / {\sim}$ is a "diamond" on $\mathbb{R}^2$
Best Answer
You are right that when people draw it as a cone, they're taking a bit of license. Identifying the circle at the end of the cylinder to a single point technically does not change the "slope" of the cylinder. A more accurate depiction would be to keep the cylinder with same slope (say, horizontal), no shrinking or tapering, and just color the boundary circle at $0$ to notate that all points on the circle are the same.
However, this is hard to understand if you're new to quotients (infinitely many colored points represent a single point??), while a cone is quite easy to understand. And as far as the topology is concerned, the two options are homeomorphic.
And the tapering or sloped cone picture shows you something: points near the apex are near each other. On the cylinder (with colored end circle) that's harder to see.
And while the tapered cone picture is not isometric to a straight cylinder with its boundary circle smashed (which is why I say they are "taking license" with this picture), they are homeomorphic. If all you care is about the topology, then it is harmless to use the easier to draw and easier to understand cone.