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Set $X=\{1,2\} \times\mathbb Z^+$ where $\mathbb Z^+$ is positive intergers. Consider $X$ under dictionary order. The order topology on $X$ is not discrete. Why it is not discrete here? Why I cannot use $\{1,2\}\times[a,b)$ form to get a basis?

MattAllegro
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user45765
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    The space ${1,2}\times\textbf{Z}_+$ is not discrete because the singleton ${(2,1)}$ is not open. Do you see why? –  May 21 '14 at 21:55
  • Related: http://math.stackexchange.com/questions/734376/show-that-the-lexicographic-order-topology-for-mathbbn-times-mathbbn-is – Asaf Karagila May 21 '14 at 22:10
  • And also: http://math.stackexchange.com/questions/638064/order-topology-on-the-set-x-1-2-times-mathbbz – Asaf Karagila May 21 '14 at 22:12

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