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Difference between revisions of "Order topology"

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Here  "open interval" means a set of the form
 
Here  "open interval" means a set of the form
 
$$
 
$$
\{ x \in X : a < x \}\,,\ \{ x \in X : x < b \}\ \text{or}\ \{ x \in X : a < x < b \}
+
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
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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
 
$$
 
$$
\{ x \in x : a \le x \le b \}
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\{ 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]]) $(x_\alpha)_{\alpha \in A}$ 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 $\sup_\alpha l_\alpha = x = \inf_\alpha u_\alpha$.
+
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====
 
====References====
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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>
 
<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>
 
</table>
 +
 +
====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.
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====References====
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<table>
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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>
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Revision as of 19:02, 20 October 2016

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
How to Cite This Entry:
Order topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order_topology&oldid=39448
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article