Difference between revisions of "Order topology"
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− | The topology < | + | The [[Topological structure (topology)|topological structure]] $\mathcal{T}_{<}$ on a [[linearly ordered set]] $X$ with linear order $<$, which has a [[base]] consisting of all possible [[Interval, open|open interval]]s of $X$. |
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====Comments==== | ====Comments==== | ||
− | Here "interval" | + | 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 set]]s 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 $\sup_\alpha l_\alpha = x = \inf_\alpha u_\alpha$. | ||
− | + | 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==== | |
+ | <table> | ||
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1973)</TD></TR> | ||
+ | <TR><TD valign="top">[a2]</TD> <TD valign="top"> O. Frink, "Topology in lattices" ''Trans. Amer. Math. Soc.'' , '''51''' (1942) pp. 569–582</TD></TR> | ||
+ | <TR><TD valign="top">[a3]</TD> <TD valign="top"> A.J. Ward, "On relations between certain intrinsic topologies in partially ordered sets" ''Proc. Cambridge Philos. Soc.'' , '''51''' (1955) pp. 254–261</TD></TR> | ||
+ | <TR><TD valign="top">[b1]</TD> <TD valign="top"> L.A. Steen, J.A. Seebach Jr., "Counterexamples in topology", 2nd ed., Springer (1978) {{ISBN|0-387-90312-7}} {{ZBL|0386.54001}}</TD></TR> | ||
+ | </table> | ||
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− | + | {{TEX|done}} | |
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Latest revision as of 16:45, 23 November 2023
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 $\sup_\alpha l_\alpha = x = \inf_\alpha u_\alpha$.
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
[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 |
[b1] | L.A. Steen, J.A. Seebach Jr., "Counterexamples in topology", 2nd ed., Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001 |
Order topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order_topology&oldid=15011