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I tried to solve that but was lost. How can i find the order topology of a finite linearly ordered topology?

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  • $\begingroup$ Do you know what is meant by the "order topology" induced by a linear order? $\endgroup$ – MPW Nov 22 '19 at 18:55
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The order topology on any linearly ordered set is always $T_1$, because the closed interval $[x, x] $ is closed for all $x$. But a finite $T_1$ space is discrete.

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  • $\begingroup$ $T_1$ is the symbol of which space? $\endgroup$ – Almaa Nov 22 '19 at 19:49
  • $\begingroup$ A $T_1$ space is a space where points are closed. $\endgroup$ – Matt Samuel Nov 22 '19 at 19:52
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    $\begingroup$ Alternatively, $\{y: y < x\}$ and $\{y ; y > x\}$ are subbasic open in the order topology, and so $\{x\}$, the complement of their union, is closed. And so all subsets of a finite space are closed etc. $\endgroup$ – Henno Brandsma Nov 22 '19 at 21:51
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Any finite ordered set is well-ordered, hence a finite ordinal, i.e. homeomorphic to a set of the form $\{0,1,\ldots,n\}$, so discrete.

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