Order topology

The topological structure $\mathcal{T}_{<}$ on a linearly ordered set $X$ with linear order $<$, which has a base consisting of all possible open intervals of $X$.

Comments

Here "open interval" means a set of the form $$ R_a = \{ x \in X : a < x \}\,,\ L_b = \{ x \in X : x < b \}\ \text{or}\ (a,b) = R_a \cap L_b = \{ x \in X : a < x < b \} $$ where $a,b$ are given elements of $X$. The order topology may be considered on partially ordered sets as well as linearly ordered sets; on a linearly ordered set it coincides with the interval topology which has the closed intervals $$ \{ x \in X : a \le x \le b \} $$ as a subbase for the closed sets, but in general it is different. On a complete linearly ordered set, the order topology is characterized by order convergence: that is, a net (see Generalized sequence) $(x_\alpha)_{\alpha \in A}$ indexed by a directed set $A$ converges to a point $x$ if and only if there exist an increasing net $l_\alpha$ and a decreasing net $u_\alpha$, indexed by the same directed set $A$, such that $l_\alpha \le x_\alpha \le u_\alpha$ for all $\alpha \in A$ and $\sup_\alpha l_\alpha = x = \inf_\alpha u_\alpha$.

References

[a1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[a2]	O. Frink, "Topology in lattices" Trans. Amer. Math. Soc. , 51 (1942) pp. 569–582
[a3]	A.J. Ward, "On relations between certain intrinsic topologies in partially ordered sets" Proc. Cambridge Philos. Soc. , 51 (1955) pp. 254–261

Comments

The left order or left interval topology is the topology with the $L_b$ as a basis for the open sets; similarly the right order topology has the $R_a$ as basis.

References

[b1]	L.A. Steen, J.A. Seebach Jr., "Counterexamples in topology", 2nd ed., Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001