# Simple Topology Equivalence Question

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1. ## Simple Topology Equivalence Question

Hello,

I have two questions about equivalence in topology.

1) Are an oval and a two dimensional double-torus equivalent?
2) Are a circle and a two dimensional donut equivalent?

Thanks,

2. i think that will depend on how you project the torus/donut into 2-space. Either way, id vote "no"

3. Good, that was the answer I hope hoping for. Actually I messed up the second question. I meant to ask
3) if a circle and a square are equalivalent.
and
4) if a pentagon and a square are equalivalent.

The premise of the first two questions was that one object contained a hole while the other didn't. I was wondering if the hole could be "closed" by enough continous stretching. I had favored no as eventually you would have to combine combine the stretched out part with the edge of hole, which I questioning the validity of that operation.

The issue with the third and fourth questions is that there is a variance in side numbers. I am thinking no for question 3 and yes for question 4, but someone clearing that up for me would be very helpful.

Thanks again,

4. i dont think i quite understand anymore...

5. Here, I will rephrase my questions and add a visualization at the end.

1) Are a rectangle and a rectangle with a hole equalivant?
2) Are an octagon and a rectangle equivilant?
3) Are a circle and a rectangle rectangle equivilant?

I hope that's understandable.

Thanks,

6. well, that all depends on how you define "equivalent". Is there more information to these questions? it all seems a little vague

7. By equivalent I meant one could be tranformed into the other using standard "rubber sheet geometry" rules like you can stretch but not tear, twist, etc. Apart from that I don't have any more information. If that isn't enough I suppose I'll track down some people from school who might be able to help me.

8. ah ok. Then id say 1 is false, 2 is true, and 3 is false... but i guess the real question is why

9. Yum...topology. I always wondered why I took that class. If I remember right, the only property that can make two shapes topologically non-equivalent is if two arbitrarily close points from shape A are not arbitrarily close in shape B. In your examples, I'd say 1) false, 2) true, and 3) true.

10. Originally Posted by Mad Cow
1) Are a rectangle and a rectangle with a hole equalivant?
2) Are an octagon and a rectangle equivilant?
3) Are a circle and a rectangle rectangle equivilant?
1) No
2) Yes
3) Yes

Originally Posted by Perspective
well, that all depends on how you define "equivalent". Is there more information to these questions? it all seems a little vague
Two sets A and B are equivalent in this sense if there exists a bijective continuous function from A to B.
Originally Posted by Perspective
but i guess the real question is why
The proof that a circle and a rectangle are topologically equivalent can be done by observing that
x = r cos t , 0<r<1
y = r sin t , 0<t<2pi
is a circle and the (r,t) space is a rectangle.