Glue both pairs of opposite edges of the square and the result is a torus, the surface of a doughnut. It is also the product of two circles, which says something worth keeping: a space built by gluing can turn out to be a space built by multiplying, and the two descriptions are equally good.

Roll the sheet into a tube, then bend the tube round and join its two ends. The order does not matter, and the reason it does not is that both pairs of edges are glued independently of each other.

The gluing diagram of a torus: the two edges marked $a$ are joined and the two marked $b$ are joined, each pair the same way up. The four corners all become a single point, since each is glued to the next round the square.

Glue $(0,y)$ to $(1,y)$ and $(x,0)$ to $(x,1)$ in the square. The result is homeomorphic to , the torus. The map wraps each coordinate round its own circle, so the two gluings happen independently — which is exactly what a product is.

• The map $(x,y)\mapsto\big(\gamma(x),\gamma(y)\big)$ with $\gamma(t)=(\cos2\pi t,\sin2\pi t)$ is continuous into the product, onto, and glues exactly the pairs the two instructions name.

Wrap each coordinate round its own circle. Both coordinates of the result are continuous, so the map into the product is continuous. It is onto, since every pair of angles occurs. Two points of the square have the same image exactly when their first coordinates agree or are the two ends, and the same for their second coordinates — which is precisely what the two gluing instructions say together. The square is compact and the product of two circles is Hausdorff, so the recognition theorem identifies the glued square with the product.

Wrap each coordinate round its own circle.

Each coordinate is continuous, so the map into the product is.

The two coordinates are glued independently, which is the pair of instructions.

A product of compact spaces is compact and a product of Hausdorff spaces is Hausdorff.

The recognition theorem finishes it, as in the two previous lessons.

Gluing both pairs of edges wraps each coordinate on its own, which is a product of circles.

• Both pairs of edges glued the same way up gives the torus.
• It is the product of two circles, and the two descriptions agree.
• All four corners become one point.
• A flip gives the Klein bottle, and the sphere is a different space again.

Great work! You now know how to:

• glue both pairs of edges and recognise the torus;
• see it as a product of two circles;
• track what happens to the four corners;
• name the Klein bottle and the sphere as different results.