Long line (topology)

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as an important counterexample in topology.^[1] Intuitively, the usual real-number line consists of a countable number of line segments $[0,1)$ laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.

^ Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. pp. 71–72. ISBN 978-0-486-68735-3. MR 0507446. Zbl 1245.54001.

[SS7172-1] Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. pp. 71–72. ISBN 978-0-486-68735-3. MR 0507446. Zbl 1245.54001.

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