Closed surfaces and closed curves

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  • Thread starter flamengo
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  • #1
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How to know if a surface is closed ? Surfaces such as x^(2/3) + y^(2/3) + z^2 = 1 or x^6 + y^6 + z^6 = 1. Is there a way to know if these surfaces are closed ( without the help of a computer program) ? Is there a general way to know if a surface is closed ? What about curves ? How to know if a curve is closed ? A curve such as x^6 + y^6 = (x^4)y. Is there a way to know if this curve is closed ? Is there a general way to know if a curve is closed ? And what subjects should I learn to know if a curve or a surface is closed ?
 

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  • #2
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How to know if a surface is closed ? Surfaces such as x^(2/3) + y^(2/3) + z^2 = 1 or x^6 + y^6 + z^6 = 1. Is there a way to know if these surfaces are closed ( without the help of a computer program) ? Is there a general way to know if a surface is closed ? What about curves ? How to know if a curve is closed ? A curve such as x^6 + y^6 = (x^4)y. Is there a way to know if this curve is closed ? Is there a general way to know if a curve is closed ? And what subjects should I learn to know if a curve or a surface is closed ?
To answer "Is it closed?" one firstly needs a topology, that defines the terms open and closed. I assume you refer to the standard Euclidean topology here. So you could either consider where the limit points of curves and surfaces have to exists, namely in those curves again, as you cannot find a sequence of points within them, that has a limit outside of them, or, which I think would be easier, consider a point outside, measure its distance to the surface, which is strictly positive, and show that the open ball with radius half this distance and center the point is completely outside the surface, which makes the set outside open and thus the surface closed.
 
  • #3
chiro
Science Advisor
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Hey flamengo.

I'd recommend you look at complex analysis [which looks at counting loops if you parameterize them with complex numbers] as well as differential geometry when it comes to curvature.

Some hints when it comes to different kinds of curvature:

https://en.wikipedia.org/wiki/Curvature
 

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