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Seifert H., Threlfall W. Topology of three-dimensional fibered areas. A textbook of topology (AP, 1980)(ISBN 0126348502)
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Additional info for Seifert and Threlfall, A textbook of topology
Euclidean 3-space closes to form the 3-sphere, when we require that conformal mappings, which are circle preserving mappings of Euclidean 3-space, (Klein [ l , Section 501; Blaschke [ I , Section 40]), be one-to-one correspondences. Among these mappings is the mapping by reciprocal radii. Under this mapping, the center point of the unit ball (the ball of inversion) 17 4. HIGHER DIMENSIONAL MANIFOLDS has no image in Euclidean 3-space. We now wish to close Euclidean 3-space in order to generate the 3-sphere by adding the image point of this one point (and not the points of a whole improper plane as we did to obtain projective 3-space).
One can prove in like manner that the boundary of (n relative to 92 is a closed subset of 5N. Let 'm, and (m, be two disjoint bounded closed sets in '8" and let d(P,P,) be the distance of a point P I of '132, to a point P, of (m,. The greatest lower bound 6 of all the distances d(P,P,) is called the distance between the sets 92, and 2R2. The greatest lower bound is adopted because (m, and (m, are closed and bounded. It is > 0 because they are disjoint. The diameter of a closed bounded set in W" will be defined to be the least upper bound of the distances of any two points of the set.
Now let 9 be a subset of (m. It is in fact true that the closed hull 9' is formed exactly by (n and its boundary, @. To demonstrate this we must show that the set union 8 9 is a closed subset of '132. In the latter case R will lie at a distance < E from R . Since there exist points of 9 which are arbitrarily close to R (from the definition of boundary points), it then follows that there will exist points of % which are at a distance < e from R . One can prove in like manner that the boundary of (n relative to 92 is a closed subset of 5N.
Seifert and Threlfall, A textbook of topology by Author Unknown