Question 1

My mathematician husband told me that Klein bottle does not have volume. But it looks that this bottle can hold liquid. How come?

Accepted Answer

Both of you are right!  Yes, the glass Klein bottle can hold water - you can put about 13 ounces "into" it.  Mathematically, as your husband points out, the bottle doesn't divide the universe into an "inside" and an "outside".  So, to a topologist, it has zero volume.  A solid ball and an empty cube both separate the universe into an inside and an outside, as does a sealed wine carafe.  Other shapes - a rectangle or a hemisphere - don't separate the universe and in this sense, do not have volume.

Having said this, I don't recommend that you fill your Klein bottle with water, mainly because it's difficult to dry.  Filling it is fairly easy, but because there's little air circulation, the last few droplets of water  are difficult to remove.  Leave it inverted overnight, and the droplets will gather in the handle, then you can shake 'em out.

Best wishes,   -Cliff Stoll

Question 2

Why is Faraday's law of induction and Gauss' law for magnetic fields included in the description?

Accepted Answer

An interesting question, worthy of a detailed explanation.  These are two of Maxwell's equations.   ∇⃗ · B⃗ = 0 is Gauss's law: there must be zero net flux of magnetic field lines out of a closed surface.  This implies that there are no magnetic monopoles.  Our Klein bottles are delivered with no magnetic monopoles - we guarantee it.  Should you find a pesky magnetic monopole, we will refund your purchase price.   ∇×E=−dB/dt , also known as Faraday's law of induction, shows that a time changing magnetic field creates an electric field.  This is valid for oriented manifolds, due to the Kelvin-Stokes theorem. It does not apply to a nonorientable manifold (such as a Klein bottle in R4-space).  Our Klein bottles are immersed (not embedded) in R3-space, and so,  ∇×E=−dB/dt  is valid locally, at least in a nonrelativistic frame of reference.  We are confident that each Acme Klein bottle will not interfere with classical electromagnetism, and so we've included this in the description.

Question 3

Is this compatible with my Klein Bottle Opener?

Accepted Answer

Hmm: A Klein bottle is a closed manifold ... by this, I mean that it has no boundary.  No edge.  It's a compact manifold without boundary.
  So, your Klein bottle opener shouldn't be able to open it ...   
If you do open the Klein bottle, I think you'll wind up with a pair of mathematically closed Mobius loops - each a Mobius loop with boundary. 
 Remove the points at the boundary, and the Mobius loop becomes mathematically open.  Like the difference between the closed unit interval  [0...1] and the open unit interval (0...1).   
All of which dodges your question ...

Question 4

Dear Cliff, why I cannot buy it in Europe?

Accepted Answer

I happily ship to Europe - over the past 25 years, I've sent Klein bottles to math folk around the globe.  However, I've had trouble writing the interface that connects my computer to the Amazon Marketplace system, especially Amazon-Europe.  I coded it for Amazon-USA, but not yet for other regions.

Product Dimensions	10 x 10 x 20 cm; 226.8 g
Item Weight	8 Ounces
Specific uses	Educational, Gift
Item volume	0.01 Microliters
Is mould resistant	Bisphenol A (BPA) Free
Finish types	Glossy
Care instructions	Microwave Safe
Batteries required	No

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