Definition:Real Number/Real Number Line
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Definition
- It can be shown (and intuitively understood) that the set of real numbers is isomorphic to any infinite straight line in space.
Thus we can identify any (either physically drawn or imagined) line with the set of real numbers and thereby illustrate truths about the real numbers by means of diagrams.
- It can be shown that the real number line is a metric space.
Hence the real number line is also a topological space.
- It can be shown that the real number line is a vector space.
Sources
- J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.1$: Example $2$
