More identification spaces

A sphere (again)

Start with a disc. Use an equivalence relation to identify points on the boundary as shown.
That is, (x₁ , y₁) ~ (x₂ , y₂) if (x₁ , y₁) = (x₂ , y₂) or
x₁ = -x₂ and y₁ = y₂ and x₁² + y₁² = 1 and x₂² + y₂² = 1.
This is like folding a piece of pastry to make a "bridie" or "pastie"


A inverse image of a neighbourhood of a point in X/~ is like an ordinary neighbourhood for an interior point and is the union of a pair of "semi-discs" for a point on the boundary.


A cylinder

Glue the ends of a strip using the same equivalence relation used to make the circle from a closed interval.
This space then gets the same topology as a subset of R³ , as a product S¹ I and as an identification space.


A Möbius band

This was invented by the German mathematician August Möbius (1790 to 1868) in 1858.

"Glue" a strip as above, but this time after a half twist.
Note that the "centre line" becomes a circle in R³.


Remark

One can make a cylinder by giving the strip a full twist before gluing. This produces something homeomorphic to the above cylinder but with an embedding in R³ in a very different way. You can see this by cutting the surface along the dotted line shown.