We're going to need some basic concepts from topology.

Intuitively speaking, topology is an abstraction of geometry that concentrates on the properties of geometric spaces that don't change when the space is subjected to certain kinds of continuous deformations.

A topological space consists of an arbitrary set X of points together with a topology (defined below) that, intuitively speaking, gives us a way of talking about which points are nearby. More formally, a topology T is a collection of subsets of X such that

- the empty set is an element of T
- X is an element of T
- intersecting any finite number of elements of T gives you an element of T
- taking the union of any number of elements of T gives you an element of T

Example:Suppose X is the set of ordered pairs <x,y> where both x and y are real numbers. For any point p in X and for any positive real number ε, let N(p,ε) be the set of points whose Euclidean distance from p is strictly less than ε. (N(p,ε) is said to be the neighborhood of radius ε around p.) Suppose T consists of every set that can be obtained by taking the union of a (possibly infinite) set of such neighborhoods. Then X with topology T is the important topological space known as Euclidean 2-space or R^{2}.

Example:Suppose X is the set of points on the surface of a globe. For any point p in X and for any positive real number ε, let N(p,ε) be the set of points whose great-circle distance from p is strictly less than ε. Suppose T consists of every set that can be obtained by taking the union of a (possibly infinite) set of such neighborhoods. Then X with topology T is another important topological space known as the 2-sphere.

If X is the set of points belonging to a topological space, and T is its topology, then the elements of T are said to be the open sets of that topological space.

If f is a mathematical function that maps one topological space into another, then f is said to be continuous if and only if: for every open set V of the second topological space, the set f^{-1}V (consisting of the points in the first space that f maps into V) is an open set of the first space. (This definition of continuous functions is more general than the usual definition, but coincides with the usual definition on familiar metric spaces such as Euclidean 2-space and the 2-sphere.)

If the continuous function f is also a one-to-one correspondence (aka bijection) between the two topological spaces, and its inverse is continuous, then f is said to be a homeomorphism, and the two spaces are said to be homeomorphic to each other.

If two topological spaces are homeomorphic, then there exists a continuous function f that converts one of those spaces into the other, while its inverse function f^{-1} reverses that conversion. It is also possible to apply the inverse function first, and to reverse that conversion using f. That means the two homeomorphic spaces are topologically equivalent, and differ only with regard to the reversible stretching and squeezing that's performed by f.

Example.A doughnut (torus) is homeomorphic to a coffee cup, but is not homeomorphic to the 2-sphere or to Euclidean 2-space.

Example.The 2-sphere is homeomorphic to the surface of a cube, but is not homeomorphic to Euclidean 2-space.

Example.If you remove any single point from the 2-sphere, the topological subspace that remains is homeomorphic to Euclidean 2-space.

That last example is so important that I'm going to define an explicit homeomorphism for it. Let's think of the 2-sphere as a perfect sphere embedded in Euclidean 3-space, and let's think of the point we remove as the south pole. For any other point p, define r(p) as the great-circle distance from p to the north pole, define s(p) as the great-circle distance from p to the south pole, and define φ(p) as the longitude of p (with any prime meridian you like); if p is the north pole, then define φ(p) to be zero.

Define f(p)=<(r(p)/s(p)) cos φ(p), (r(p)/s(p)) sin φ(p)>. Then f(p) defines a one-to-one correspondence between the 2-sphere without its south pole and Euclidean 2-space. It's easy to check that f(p) is continuous and has a continuous inverse. Euclidean 2-space is therefore homeomorphic to the 2-sphere without its south pole.

Instead of omitting just the south pole, we could omit any closed disk that's centered on the south pole and define s(p) to be the great-circle distance from p to the circular boundary of that missing disk. With that modification, the definition of f(p) above would give us a homeomorphism between Euclidean 2-space and the 2-sphere without that closed disk.

Topologically speaking, it doesn't matter whether you remove a single point from the 2-sphere or an entire closed disk. Either way, you get a topological space that's equivalent to Euclidean 2-space.

In like manner, *removing a single point from Euclidean 3-space gives you a topological space that's homeomorphic to what you get by removing an entire closed ball from Euclidean 3-space.*

That italicized fact is the topological fact we need to explain where Abrams went wrong. Later on, we will also use the fact that the 2-sphere is *not* homeomorphic to Euclidean 2-space.