Now for the confusing part. Instead of calling the spaces isomorphic, as we do with groups, rings, fields, modules, etc, the spaces are called homeomorphic, and the function is called a homeomorphism. Don't ask me why; I guess the topologists wanted their own special words.
If S is homeomorphic to T is homeomorphic to U, then S is homeomorphic to U. Simply compose the two functions. Thus homeomorphic spaces form equivalence classes in the universe of topological spaces. Just as isomorphic groups are sometimes considered to be the same "group", so homeomorphic spaces are sometimes considered to be the same "space". We're just relabeling points, without changing the topology, and that's not very interesting. Let's consider an example.
The punctured sphere is the same space as the plane. Pull the sphere apart and lay it flat, and spread it out to infinity. If you want to be analytic, do the following. Place a sphere on the xy plane, its south pole at the origin. Delete the north pole; that's the point that is missing. Now draw a ray from the north pole to any point in the xy plane. The ray intersects the sphere in exactly one point. This is the 1-1 map equating the punctured sphere and the plane, and if you write it as a formula, it is an algebraic expression employing trig functions, and is bicontinuous, hence it is a homeomorphism.
In another example, center the sphere at the origin and remove the north and south pole. Enclose the sphere in an infinite cylinder, the points that are 1 unit away from the z axis. Now draw rays from the origin to the cylinder. These rays map the cylinder onto the sphere, and vice versa. The map is bicontinuous, and the spaces are homeomorphic.