# What is the order topology on a finite linearly ordered set?

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

• Do you know what is meant by the "order topology" induced by a linear order? – MPW Nov 22 '19 at 18:55

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.
• $T_1$ is the symbol of which space? – Almaa Nov 22 '19 at 19:49
• A $T_1$ space is a space where points are closed. – Matt Samuel Nov 22 '19 at 19:52
• 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. – Henno Brandsma Nov 22 '19 at 21:51
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.