In the book "Basic Topology" by M.A.Armstrong I found an explanation about how to construct a Klein bottle. I had to reread it 5 times and I was still not quite convinced. I am retelling it here.

I was scratching my head around how this procedure actually produces a Klein bottle until I found this question in

Klein-bottle-as-two-Möbius-strips

This picture there in one of the answers is really really nice, it really shows what happens if we cut a Klein bottle in half - we really get two Möbius strips as a result. The cut is done by a plane "parallel to the handle" which cuts the bottle in two symmetric parts.

So... it's really for a reason they say "a picture is worth a thousand words".

*Begin with a sphere, remove two discs from it, and add a Möbius strip in their places.**A Möbius strip has after all a single circle as boundary, and all that we are asking**is that the points of its boundary circle be identified with those of the boundary**circle of the hole in the sphere. One must imagine this identification taking place**in some space where there is plenty of room (euclidean four-dimensional space will do).**This cannot be realized in three dimensions without having each Möbius strip**intersect itself. The resulting closed surface is called the Klein bottle.*I was scratching my head around how this procedure actually produces a Klein bottle until I found this question in

**MathSE**.Klein-bottle-as-two-Möbius-strips

This picture there in one of the answers is really really nice, it really shows what happens if we cut a Klein bottle in half - we really get two Möbius strips as a result. The cut is done by a plane "parallel to the handle" which cuts the bottle in two symmetric parts.

So... it's really for a reason they say "a picture is worth a thousand words".

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