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I thought it might be relevant to describe the topology of the real line without any use of the Euclidean metric or any other metric for that matter and without any algebraic structure.
A little web research revealed several ways to characterize the topology of the real line or equivalently any open interval on the real line. The reference is this post on Mathoverflow
https://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line
The easiest one for me to understand was
- The topological real line is charactrized as a connected locally connected separable regular topological space in which the complement of any point is two disjoint connected sets. The quoted references for this description are
"The topological characterization of an open linear interval", Proc. London Math. Soc.(2) 41 (1936), 191-198
"On the topological characterization of the real line", Department of Mathematics, Carnegie-Mellon University, Report #69-36, 1969
A relevant point made in the post was that the first reference did not say regular space but instead said metric space. Regular is less restrictive than metric space which shows that a metric is not required to define the topology of the real line.
A little web research revealed several ways to characterize the topology of the real line or equivalently any open interval on the real line. The reference is this post on Mathoverflow
https://mathoverflow.net/questions/76134/topological-characterisation-of-the-real-line
The easiest one for me to understand was
- The topological real line is charactrized as a connected locally connected separable regular topological space in which the complement of any point is two disjoint connected sets. The quoted references for this description are
"The topological characterization of an open linear interval", Proc. London Math. Soc.(2) 41 (1936), 191-198
"On the topological characterization of the real line", Department of Mathematics, Carnegie-Mellon University, Report #69-36, 1969
A relevant point made in the post was that the first reference did not say regular space but instead said metric space. Regular is less restrictive than metric space which shows that a metric is not required to define the topology of the real line.
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