Projectively extended real line: Difference between revisions

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The projectively extended real line can be visualized as the real number line wrapped around a [[circle]] (by some form of [[stereographic projection]]) with an additional [[point at infinity]].]]

In [[real analysis]], the '''projectively extended real line''' (also called the [[one-point compactification]] of the [[real line]]), is the extension of the [[set (mathematics)|set]] of the [[real number]]s, <math>\mathbb{R}</math>, by a point denoted {{math|∞}}.<ref name=":0">{{Cite book |last=NBU |first=DDE |url=https://books.google.com/books?id=4i7eDwAAQBAJ&dq=%22Projectively+extended+real+line%22+-wikipedia&pg=PA62 |title=PG MTM 201 B1 |date=2019-11-05 |publisher=Directorate of Distance Education, University of North Bengal |language=en}}</ref> It is thus the set <math>\mathbb{R}\cup\{\infty\}</math> with the standard arithmetic operations extended where possible,<ref name=":0" /> and is sometimes denoted by <math>\mathbb{R}^*</math><ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |title=Projectively Extended Real Numbers |url=https://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html |access-date=20232025-0107-2213 |website=mathworld.wolfram.com |language=en}}</ref> or <math>\widehat{\mathbb{R}}.</math> The added point is called the [[point at infinity]], because it is considered as a neighbour of both [[End (topology)|ends]] of the real line. More precisely, the point at infinity is the [[limit of a sequence|limit]] of every [[sequence]] of real numbers whose [[absolute value]]s are [[Sequence#Increasing and decreasing|increasing]] and [[bounded function|unbounded]].

The projectively extended real line may be identified with a [[real projective line]] in which three points have been assigned the specific values {{math|0}}, {{math|1}} and {{math|∞}}. The projectively extended real number line is distinct from the [[affinely extended real number line]], in which {{math|+∞}} and {{math|−∞}} are distinct.

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a \cdot \infty & = \frac{a}{0} \\

\end{align}</math>

The following is true whenever theexpressions right-handinvolved side isare defined, for any <math>a, b, c \in \widehat{\mathbb{R}}.</math>

:<math>

\begin{align}

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Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},</math> <math>p \in \widehat{\mathbb{R}},</math> and <math>L \in \widehat{\mathbb{R}}</math>.

The [[limit of a function|limit]] of ''f''&{{hairsp;}}(''x'') as {{math|''x''}} approaches ''p'' is ''L'', denoted

: <math>\lim_{x \to p}{f(x)} = L</math>

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

The [[one-sided limit]] of ''f''&{{hairsp;}}(''x'') as ''x'' approaches ''p'' from the right (left) is ''L'', denoted

: <math>\lim_{x \to p^{+}}{f(x)} = L \qquad \left( \lim_{x \to p^{-}}{f(x)} = L \right),</math>

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

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Let <math>A \subseteq \widehat{\mathbb{R}}</math>. Then ''p'' is a [[limit point]] of ''A'' if and only if every neighbourhood of ''p'' includes a point <math>y \in A</math> such that <math>y \neq p.</math>

Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}</math>, ''p'' a limit point of ''A''. The limit of ''f''&{{hairsp;}}(''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 <math>x \in A \cap C</math> implies <math>f(x) \in B.</math>

This corresponds to the regular [[continuity (topology)|topological definition of continuity]], applied to the [[subspace topology]] on <math>A\cup \lbrace p \rbrace,</math> and the [[restriction (mathematics)|restriction]] of ''f'' to <math>A \cup \lbrace p \rbrace.</math>