Long line (topology)

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In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain sense "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. Intuitively, the usual real-number line consists of a countable number of line segments [ 0 , 1 ) {\displaystyle [0,1)} laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.

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