How to draw $S^1\times I$

Question

I always thought $P=S^1\times I$, where $S^1$ is the circle and $I=(0,1)$ with the standard topology is the surface of the cylinder, but I was reading a book which says me another thing:

Even, if the book is right, I didn't understand why $P$ is the open annulus, because the radius inside the annulus is not the same as $S^1$.

So, why $P$ is not the surface of the cylinder? why $P$ is the open annulus? what the author said doesn't convince me.

Thanks

Topologically, you cannot "see" the radius. So to go from one to the other, just "push" the middle out to make a cylinder. — Tobias Kildetoft, Sep 23 '14 at 20:25
When you say you think it's the surface of a cylinder, do you think it includes the discs at each end, or just the tube? — MartianInvader, Sep 23 '14 at 20:26
So what happens if you shrink the top end of the tube a little, then expand the bottom part a little (so it's kind of a cut-off cone), then smash the whole thing flat into the floor? — MartianInvader, Sep 23 '14 at 20:29

score 5 · Answer 1 · answered Sep 23 '14 at 21:40

5

Let $C$ be the cylinder consisting of the set of points in 3-space with cylindrical coordinates $(r,\theta,z)$ such that $r=1$ and $0 \lt z \lt 1$.

Let $A$ be a particular example of an annulus $-$ say, consisting of the set of points in the plane with polar coordinates $(r,\theta)$ such that $1 \lt r \lt 2$.

The cylinder $C$ is probably what you have in mind when you think of $S^1 \times (0,1)$. But the function which maps $(1,\theta,z) \in C$ to $(z+1,\theta) \in A$ is a homeomorphism. So the topological spaces $C$ and $A$ are both the same as the space $S^1 \times (0,1)$ (where "the same as" means "homeomorphic to").

answered Sep 23 '14 at 21:40

echinodermata

4,219

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Why $(z+1,\theta)$? – user42912 Sep 24 '14 at 06:02
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@user42912 The goal was to show that $C$ and $A$ are homeomorphic; the direct way to do this is to given an example of a continuous function $f:C \rightarrow A$ whose inverse is also a continuous function. The function $f(1,\theta,z)=(z+1,\theta)$ (using cylindrical and polar coordinates, respectively), is just an example; I chose "$z+1$" since $C$ has $0 \lt z \lt 1$ while $A$ has $1 \lt r \lt 2$. – echinodermata Sep 25 '14 at 01:02

How to draw $S^1\times I$

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