Previous page (The identification topology) | Contents | Next page (Separation axioms) |

**A sphere** (again)

Start with a disc. Use an equivalence relation to identify points on the boundary as shown.

That is, (*x*_{1} , *y*_{1}) ~ (*x*_{2} , *y*_{2}) if (*x*_{1} , *y*_{1}) = (*x*_{2} , *y*_{2}) or

*x*_{1} = -*x*_{2} and *y*_{1} = *y*_{2} and *x*_{1}^{2} + *y*_{1}^{2} = 1 and *x*_{2}^{2} + *y*_{2}^{2} = 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 **R**^{3} , as a product *S*^{1} *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 **R**^{3}.

**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 **R**^{3} in a very different way. You can see this by cutting the surface along the dotted line shown.

Previous page (The identification topology) | Contents | Next page (Separation axioms) |