How do we define a 4D object like a Klein bottle in 3D topology?

 

And does such matter exist in space?

Klein bottle is a mathematical object. It cannot be made physically.

Klein bottle is a mathematical object. It cannot be made physically.

... and yet, I can buy one if I want to: http://www.kleinbottle.com/

... and yet, I can buy one if I want to: http://www.kleinbottle.com/

You can buy a very nice representation of one - but as I learnt on my early days here - you cannot buy an actual klein bottle as they do not intersect with themselves which requires 4 dimensions as I strongly suspect you know.

 

That said I would heartily recommend buying one - they are really neat. And the website even gives a nice explanation of why what you are getting is 2d surfaces embedded in 3d space rather than the singular 2d surface embedded in 4d space which is a true klein bottle

 

http://www.kleinbottle.com/whats_a_klein_bottle.htm

You can buy a very nice representation of one - but as I learnt on my early days here - you cannot buy an actual klein bottle as they do not intersect with themselves which requires 4 dimensions as I strongly suspect you know.

 

 

I suspect you mean that a Klein bottle cannot be embedded in manifold of less than 4 dimensions. Maybe we should call the Klien bottle a Klein manifold to have consistent language. But in any case, the 3D space we've come to know and love is (at least locally) oriented. A left glove cannot be locally turned into a right glove.

 

But what if the 3-space is itself non-oriented? Can a Klein manifold be embedded in a 3 dimensional manifold that is itself non-oriented?

 

It occurs to me that the non-oriented Klein manifold can be embedded in itself by identity. Adding one more dimension doesn't nullify the embedding. So it does seem possible to do so in specific cases.