Real projective line

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The real projective line can be thought of as a line whose endpoints meet at infinity.

In real analysis, the real projective line (also called the one-point compactification of the real line, or the projectively extended real numbers), is the set \mathbb{R}\cup\{\infty\}, also denoted by \widehat{\mathbb{R}} and by \mathbb{R}\mathrm{P}^1.

The symbol \infty represents the point at infinity, an idealized point that bridges the two "ends" of the real line.

Dividing by zero[edit]

Unlike most mathematical models of the intuitive concept of 'number', this structure allows division by zero:

\frac{a}{0} = \infty

for nonzero a. This structure, however, is not a field, and division does not retain its original algebraic meaning in it. The geometric interpretation is this: a vertical line has infinite slope.

Extensions of the real line[edit]

The real projective line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally \infty.

Compare the extended real number line (also called the two-point compactification of the real line), which does distinguish between +\infty and -\infty.

Order[edit]

The order relation cannot be extended to \widehat{\mathbb{R}} in a meaningful way. Given a real number a, there is no convincing reason to decide that a > \infty or that a < \infty. Since \infty can't be compared with any of the other elements, there's no point in using this relation at all. As another example, both  3 < 4 and 3 > 4 are true. However, order is used to make definitions in \widehat{\mathbb{R}} that are based on the properties of reals.

Geometry[edit]

Fundamental to the idea that ∞ is a point no different from any other is the way the real projective line is a homogeneous space, in fact homeomorphic to a circle. For example the general linear group of 2×2 real invertible matrices has a transitive action on it. The group action may be expressed by Möbius transformations, (also called linear fractional transformations), with 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 R2.

Arithmetic operations[edit]

Motivation for arithmetic operations[edit]

The arithmetic operations in this space are an extension of the same operations on reals. The motivation for the new definitions is the limits of functions of real numbers.

Arithmetic operations which are defined[edit]


\begin{align} \\
a + \infty = \infty + a & = \infty, & a \in \mathbb{R} \\
a - \infty = \infty - a & = \infty, & a \in \mathbb{R} \\
a \cdot \infty = \infty \cdot a & = \infty, & a \in \mathbb{R}, a \neq 0 \\
\infty \cdot \infty & = \infty \\
\frac{a}{\infty} & = 0, & a \in \mathbb{R} \\
\frac{\infty}{a} & = \infty, &  a \in \mathbb{R} \\
\frac{a}{0} & = \infty, & a \in \mathbb{R}, a \neq 0
\end{align}

Arithmetic operations which are left undefined[edit]

The following cannot be motivated by considering limits of real functions, and any definition of them would require us to give up additional algebraic properties. Therefore, they are left undefined:


\begin{align}
& \infty + \infty \\
& \infty - \infty \\
& \infty \cdot 0 \\
& 0 \cdot \infty \\
& \frac{\infty}{\infty} \\
& \frac{0}{0}
\end{align}

Algebraic properties[edit]

The following equalities mean: Either both sides are undefined, or both sides are defined and equal. This is true for any a, b, c \in \widehat{\mathbb{R}}.


\begin{align}
(a + b) + c & = a + (b + c) \\
a + b & = b + a \\
(a \cdot b) \cdot c & = a \cdot (b \cdot c) \\
a \cdot b & = b \cdot a \\
a \cdot \infty & = \frac{a}{0} \\
\end{align}

The following is true whenever the right-hand side is defined, for any a, b, c \in \widehat{\mathbb{R}}.


\begin{align}
a \cdot (b + c) & = a \cdot b + a \cdot c \\
a & = (\frac{a}{b}) \cdot b & = \,\,& \frac{(a \cdot b)}{b} \\
a & = (a + b) - b & = \,\,& (a - b) + b
\end{align}

In general, all laws of arithmetic are valid as long as all the occurring expressions are defined.

Intervals and topology[edit]

The concept of an interval can be extended to \widehat{\mathbb{R}}. However, since it is an unordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that a, b \in \mathbb{R}, a < b):


\begin{align}
\left[a, a\right] & = \lbrace a \rbrace  \\
\left[a, b\right] & = \lbrace x \vert x \in \mathbb{R}, a \leq x \leq b \rbrace  \\
\left[a, \infty\right] & = \lbrace x \vert x \in \mathbb{R}, a \leq x \rbrace  \cup \lbrace \infty \rbrace  \\
\left[b, a\right] & = \lbrace x \vert x \in \mathbb{R}, b \leq x \rbrace  \cup \lbrace \infty \rbrace \cup \lbrace x \vert x \in \mathbb{R}, x \leq a \rbrace  \\
\left[\infty, a\right] & = \lbrace \infty \rbrace  \cup \lbrace x \vert x \in \mathbb{R}, x \leq a \rbrace  \\
\left[\infty, \infty\right] & = \lbrace \infty \rbrace 
\end{align}

The corresponding open and half-open intervals are obtained by removing the endpoints.

\widehat{\mathbb{R}} itself is also an interval, but cannot be represented with this bracket notation.

The open intervals as base define a topology on \widehat{\mathbb{R}}. Sufficient for a base are the finite open intervals and the intervals (b, a) = \{x | x \in \mathbb{R}, b < x\} \cup \{\infty\} \cup \{x | x \in \mathbb{R}, x < a\}.

As said, the topology is homeomorphic to a circle. Thus it is metrizable corresponding (for a given homeomorphism) to the ordinary metric on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on R.

Interval arithmetic[edit]

Interval arithmetic is trickier in \widehat{\mathbb{R}} than in \mathbb{R}. However, the result of an arithmetic operation on intervals is always an interval. In particular, we have, for every a, b \in \widehat{\mathbb{R}}:

x \in [a, b] \iff \frac{1}{x} \in \left [\frac{1}{b}, \frac{1}{a}\right ]

which is true even when the intervals involved include 0.

Calculus[edit]

The tools of calculus can be used to analyze functions of \widehat{\mathbb{R}}. The definitions are motivated by the topology of this space.

Neighbourhoods[edit]

Let x \in \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}.

  • A is a neighbourhood of x, if and only if A contains an open interval B and x \in B.
  • A is a right-sided neighbourhood of x, if and only if there is y \in \widehat{\mathbb{R}}, y > x such that A contains [x, y).
  • A is a left-sided neighbourhood of x, if and only if there is y \in \widehat{\mathbb{R}}, y < x such that A contains (y, x].
  • A is a (right-sided, left-sided) punctured neighbourhood of x, if and only if there is B \subseteq \widehat{\mathbb{R}} such that B is a (right-sided, left-sided) neighbourhood of x, and A = B \setminus \{x\}.

Limits[edit]

Basic definitions of limits[edit]

Let f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}.

The limit of f(x) as x approaches p is L, denoted

\lim_{x \to p}{f(x)} = L

if and only if for every neighbourhood A of L, there is a punctured neighbourhood B of p, such that x \in B implies f(x) \in A.

The one-sided limit of f(x) as x approaches p from the right (left) is L, denoted

\lim_{x \to p^{+}}{f(x)} = L    \left (\lim_{x \to p^{-}}{f(x)} = L\right )

if and only if for every neighbourhood A of L, there is a right-sided (left-sided) punctured neighbourhood B of p, such that x \in B implies f(x) \in A.

It can be shown that \lim_{x \to p}{f(x)} = L if and only if both \lim_{x \to p^{+}}{f(x)} = L and \lim_{x \to p^{-}}{f(x)} = L.

Comparison with limits in \mathbb{R}[edit]

The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, p, L \in \mathbb{R}, the first limit is as defined above, and the second limit is in the usual sense:

  • \lim_{x \to p}{f(x)} = L is equivalent to \lim_{x \to p}{f(x)} = L.
  • \lim_{x \to \infty^{+}}{f(x)} = L is equivalent to \lim_{x \to -\infty}{f(x)} = L.
  • \lim_{x \to \infty^{-}}{f(x)} = L is equivalent to \lim_{x \to +\infty}{f(x)} = L.
  • \lim_{x \to p}{f(x)} = \infty is equivalent to \lim_{x \to p}{|f(x)|} = +\infty.
  • \lim_{x \to \infty^{+}}{f(x)} = \infty is equivalent to \lim_{x \to -\infty}{|f(x)|} = +\infty.
  • \lim_{x \to \infty^{-}}{f(x)} = \infty is equivalent to \lim_{x \to +\infty}{|f(x)|} = +\infty.

Extended definition of limits[edit]

Let A \subseteq \widehat{\mathbb{R}}. Then p is a limit point of A if and only if every neighbourhood of p includes a point y \in A such that y \neq x.

Let f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}, p a limit point of A. The limit of f(x) as x approaches p through A is L, if and only if for every neighbourhood B of L, there is a punctured neighbourhood C of p, such that x \in A \cap C implies f(x) \in B.

This corresponds to the regular topological definition of continuity, applied to the subspace topology on A\cup \lbrace p \rbrace, and the restriction of f to A\cup \lbrace p \rbrace.

Continuity[edit]

Let

f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},\quad p \in \widehat{\mathbb{R}}.

f is continuous at p if and only if f is defined at p and:

\lim_{x \to p}{f(x)} = f(p).

Let

f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},\quad A \subseteq \widehat{\mathbb{R}}.

f is continuous in A if and only if for every p \in A, f is defined at p and the limit of f(x) as x approaches p through A is f(p).

An interesting feature is that every rational function P(x)/Q(x), where P(x) and Q(x) have no common factor, is continuous in \widehat{\mathbb{R}}. Also, If tan is extended so that

\tan\left(\frac{\pi}{2} + n\pi\right) = \infty\text{ for }n \in \mathbb{Z},

then tan is continuous in \mathbb{R}. However, many elementary functions, such as trigonometric and exponential functions, are discontinuous at \infty. For example, sin is continuous in \mathbb{R} but discontinuous at \infty.

Thus 1/x is continuous on \widehat{\mathbb{R}} but not on the affinely extended real number system R. Conversely, the function arctan can be extended continuously on R, but not on \widehat{\mathbb{R}}.

Hyperbolic involutions[edit]

An involution is hyperbolic when it has two fixed points. There are two elementary, arithmetic, hyperbolic involutions on the real projective line: negation and reciprocation. Indeed, 0 and ∞ are fixed under negation, while 1 and −1 are fixed under reciprocation.

See also[edit]

External links[edit]