How to prove a cone $K$ is closed? By showing that its complement is open?
2026-04-03 05:52:45.1775195565
How to prove that a cone is closed?
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The statement is false.
For example, the set $$X = \{0\} \cup \{ t_1x + t_2x_2 : t_1,t_2 >0, x_1 \neq x_2 \}$$
is a cone, but if we select $y_n = \frac{1}{n}x_1 + x_2$ then notice $\lim y_n = x_2 \notin X$. The situation can be reformuated with $X - \{ 0 \}$ depending on your definition of a cone.