compact torus?

**Mauritzvdworm** · August 24th 2009, 08:18 AM

let $T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}$

can we show that $T^{2}$ is compact?

**nirax** · August 24th 2009, 10:30 AM

Originally Posted by Mauritzvdworm

let $T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}$

can we show that $T^{2}$ is compact?

embedd $T^{2}$ inside $\mathcal{R}^{3}$ ... easy to construct such a map. then you can clearly see that it is a closed and bounded subset of $\mathcal{R}^{3}$ , hence by heine-borel theorem it is compact

**Mauritzvdworm** · August 24th 2009, 11:06 AM

could you give me an example of such a map?

**nirax** · August 24th 2009, 11:21 AM

i will describe it by words, you write down the expression. call the parameters in the domain a and b. call the corordinate of R^3 as x, y, z.

in the x-z plane draw a circle of radius 2. call this circle C. parametrize its points by a, which is to say that you choose a reference line segment radius. consider a plane perpendicular to x-z plane, passing through y-axis, such that its intersection with x-z plane makes an angle of a with the refernce radius. this plane rotates along the y-axis, the angle of rotation being a.

when you have rotated this plane by angle a, draw a circle of radius 1 around the point (2cos a, 0, 2sin a) in this plane. parametrize the points of this circle by b ...

try to visualize what i am saying ... it is really easy if seen pictotrially.
i think u can fill in the details. ... convince yourself that this map is smooth embedding.

**Opalg** · August 24th 2009, 12:05 PM

Originally Posted by Mauritzvdworm

let $T^{2}=\mathcal{R}^{2}/2\pi\mathcal{Z}^{2}$

can we show that $T^{2}$ is compact?

You could use the homeomorphism $T^2\cong T\times T$ , where $T = \mathbb{R}/2\pi\mathbb{Z}$ , together with the fact that a product of compact spaces is compact.

**Mauritzvdworm** · August 24th 2009, 12:13 PM

Originally Posted by nirax

this plane rotates along the y-axis, the angle of rotation being a.

does the plane rotate about the y-axis or about a point of the circle C which is in the plane with angle a with respect to the reference radius?

**nirax** · August 24th 2009, 09:00 PM

about the y-axis. ok i will give you the final answer. try to work it out. and also if you have access to 3d plots, try to plot it. maybe gnuplot cud help you

$(a,b) \longmapsto (2(1/2 +\cos{b})\cos{a}, \sin{b}, 2(1/2+\cos{b})\sin{a})$

edit :: i am not very sure i have done the computations correctly. plz check it.

**nirax** · August 24th 2009, 09:04 PM

is there a list where i can see all the mathematical fonts ?

**Chris L T521** · August 24th 2009, 09:06 PM

Originally Posted by nirax

is there a list where i can see all the mathematical fonts ?

See the LaTeX help section of the forums (in particular, the tutorials).

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