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The topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700801.png" /> on a linearly ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700802.png" />, with linear order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700803.png" />, which has a base consisting of all possible intervals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700804.png" />.
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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 $X$.
 
 
 
 
  
 
====Comments====
 
====Comments====
Here  "interval" is used in the sense of  "open interval" , i.e. a set of the form
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Here  "open interval" means a set of the form
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$$
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R_a = \{ x \in X : a < x \}\,,\ L_b = \{ x \in X : x < b \}\ \text{or}\ (a,b) = R_a \cap L_b = \{ x \in X : a < x < b \}
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$$
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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
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$$
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\{ x \in X : a \le x \le b \}
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$$
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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 $\sup_\alpha l_\alpha = x = \inf_\alpha u_\alpha$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700805.png" /></td> </tr></table>
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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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700806.png" /> (or possibly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700807.png" /> and/or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700808.png" />). The order topology may be considered on partially ordered as well as linearly ordered sets; on a linearly ordered set it coincides with the interval topology which has the closed intervals
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birkhoff, "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc. (1973)</TD></TR>
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<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>
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<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>
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<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>
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</table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o0700809.png" /></td> </tr></table>
 
  
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|Generalized sequence]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008010.png" /> converges to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008011.png" /> if and only if there are an increasing net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008012.png" /> and a decreasing net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008013.png" />, indexed by the same directed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008014.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008015.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070080/o07008017.png" />.
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{{TEX|done}}
 
 
====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></table>
 

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
How to Cite This Entry:
Order topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order_topology&oldid=15011
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article