the understanding that when the denominator of the linear fractional transformation is 0, the image is ∞. The detailed analysis of the action shows that for any three distinct points P, Q and R, there is a linear fractional transformation taking P to 0, Q to 1, and R to ∞ that is, the group of linear fractional transformations is triply transitive on the real projective line. This cannot be extended to 4-tuples of points, because the cross-ratio is invariant. The terminology projective line is appropriate, because the points are in 1-to-1 correspondence with one-dimensional linear subspaces of . Arithmetic operations Motivation for arithmetic operations The arithmetic operations on this space are an extension of the same operations on reals. A motivation for the new definitions is the limits of functions of real numbers. Arithmetic operations that are defined In addition to the standard operations on the subset of , the following operations are defined for , with exceptions as indicated: Arithmetic operations that are left undefined The following expressions cannot be motivated by considering limits of real functions, and no definition of them allows the statement of the standard algebraic properties to be retained unchanged in form for all defined cases. Consequently, they are left undefined: The exponential function cannot be extended to . Algebraic properties The following equalities mean: Either both sides are undefined, or both sides are defined and equal. This is true for any The following is true whenever the right-hand side is defined, for any In general, all laws of arithmetic that are valid for are also valid for whenever all the occurring expressions are defined. Intervals and topology The concept of an interval can be extended to . However, since it is not an ordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that ): With the exception of when the end-points are equal, the corresponding open and half-open intervals are defined by removing the respective endpoints. This redefinition is useful in interval arithmetic when dividing by an interval containing 0. and the empty set are also intervals, as is excluding any single point. The open intervals as a base define a topology on . Sufficient for a base are the bounded open intervals in and the intervals for all such that As said, the topology is homeomorphic to a circle. Thus it