Difference between revisions of "Order topology"
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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" />. | + | 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" />. |
Revision as of 19:24, 19 October 2016
The topology 
on a linearly ordered set 
, with linear order 
, which has a base consisting of all possible intervals of 
.
Comments
Here "interval" is used in the sense of "open interval" , i.e. a set of the form
 |
where 
(or possibly 
and/or 
). 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
 |
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) 
converges to a point 
if and only if there are an increasing net 
and a decreasing net 
, indexed by the same directed set 
, such that 
for all 
and 
.
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 |
Order topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Order_topology&oldid=39445