For instance, let’s say you have a pentagon and two of the vertices are at (0,0) and (0,2). What do you do to shrink it?

]]>All the best,

Charlie Sitler

]]>(1) The long line is not metrizable, and

(2) The Birkhoff-Kakutani theorem, which says that a topological group that is Hausdorff and first countable is metrizable.

Terry Tao has a nice exposition of the Birkhoff-Kakutani theorem https://terrytao.wordpress.com/2011/05/17/the-birkhoff-kakutani-theorem/

It is well-known that the long line is not metrizable. A simple proof is given in this ancient paper of mine; it shows that the first uncountable ordinal is not metrizable:

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