Editing Projectively extended real line (section)

== Intervals and topology ==
The concept of an [[interval (mathematics)|interval]] can be extended to <math>\widehat{\mathbb{R}}</math>. 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 
<math>a, b \in \mathbb{R}, a < b</math>):<ref name=":1" />{{Additional citations needed|date=January 2023}}

: <math>\begin{align}
\left[a, b\right] & = \lbrace x \mid x \in \mathbb{R}, a \leq x \leq b \rbrace  \\
\left[a, \infty\right] & = \lbrace x \mid x \in \mathbb{R}, a \leq x \rbrace  \cup \lbrace \infty \rbrace  \\
\left[b, a\right] & = \lbrace x \mid x \in \mathbb{R}, b \leq x \rbrace  \cup \lbrace \infty \rbrace \cup \lbrace x \mid x \in \mathbb{R}, x \leq a \rbrace  \\
\left[\infty, a\right] & = \lbrace \infty \rbrace \cup \lbrace x \mid x \in \mathbb{R}, x \leq a \rbrace  \\
\left[a, a\right] & = \{ a \}  \\
\left[\infty, \infty\right] & = \lbrace \infty \rbrace 
\end{align}</math>

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.<ref name=":1" />

<math>\widehat{\mathbb{R}}</math> and the [[empty set]] are also intervals, as is <math>\widehat{\mathbb{R}}</math> excluding any single point.{{efn|If consistency of complementation is required, such that <math>[a,b]^\complement = (b,a)</math> and <math>(a,b]^\complement = (b,a]</math> for all <math>a, b \in \widehat{\mathbb{R}}</math> (where the interval on either side is defined), all intervals excluding <math>\varnothing</math> and <math>\widehat{\mathbb{R}}</math> may be naturally represented using this notation, with <math>(a,a)</math> being interpreted as <math>\widehat{\mathbb{R}}\setminus \{ a \}</math>, and half-open intervals with equal endpoints, e.g. <math>(a,a]</math>, remaining undefined.}}

The open intervals as a [[base (topology)|base]] define a [[topological space|topology]] on <math>\widehat{\mathbb{R}}</math>. Sufficient for a base are the [[bounded interval|bounded]] open intervals in <math>\mathbb{R}</math> and the intervals <math>(b, a) = \{x \mid x \in \mathbb{R}, b < x\} \cup \{\infty\} \cup \{x \mid x \in \mathbb{R}, x < a\}</math> for all <math>a, b \in \mathbb{R}</math> such that <math>a < b.</math>

As said, the topology is [[homeomorphic]] to a circle. Thus it is [[metrizable]] corresponding (for a given homeomorphism) to the ordinary [[metric (mathematics)|metric]] on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on <math>\mathbb{R}.</math>