Metric and Topological Spaces

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## 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.

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JOC February 2004