# topology

## Klein Bottle

A Klein bottle is a surface with a very strange property. A surface is any object that is locally 2-dimensional;...

## Projective Planes

Most of you know how to make a Mobius band—take a strip of paper and glue the ends with a...

## Sphere Eversions

If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it...

Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and...

Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber....

## Rubber Bands Stuck on a Torus

Consider a rubber tire (a torus) with a hole in it. Suppose that there is a green rubber band stuck...

## Connected Sums

A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,...

## Borsuk-Ulam Theorem

The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given...