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单词 OrderTopology
释义

order topology


Let (X,) be a linearly ordered set. The order topology on X is defined to be the topologyMathworldPlanetmath 𝒯 generated by the subbasis consisting of open rays, that is sets of the form

(x,)={yX|y>x}
(-,x)={yX|y<x},

for some xX.

This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to saying that 𝒯 is generated by the basis of open intervals; that is, the open rays as defined above, together with sets of the form

(x,y)={zX|x<z<y}

for some x,yX.

The standard topologies on , and are the same as the order topologies on these sets.

If Y is a subset of X, then Y is a linearly ordered set under the induced order from X. Therefore, Y has an order topology 𝒮 defined by this orderingMathworldPlanetmath, the induced order topology. Moreover, Y has a subspace topology 𝒯 which it inherits as a subspaceMathworldPlanetmath of the topological space X. The subspace topology is always finer than the induced order topology, but they are not in general the same.

For example, consider the subset Y={-1}{1nn}. Under the subspace topology, the singleton set {-1} is open in Y, but under the order topology on Y,any open set containing -1 must contain all but finitely many members of the space.

A chain X under the order topology is HausdorffPlanetmathPlanetmath: pick any two distinct points x,yX; without loss of generality, say x<y. If there is a z such that x<z<y, then (-,z) and (z,) are disjoint open sets separating x and y. If no z were between x and y, then (-,y) and (x,) are disjoint open sets separating x and y.

Titleorder topology
Canonical nameOrderTopology
Date of creation2013-03-22 12:10:34
Last modified on2013-03-22 12:10:34
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id10
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 54B99
Classificationmsc 06F30
Synonyminduced order topology
Related topicOrderedSpace
Related topicLinearContinuum
Related topicProofOfGeneralizedIntermediateValueTheorem
Related topicASpaceIsConnectedUnderTheOrderedTopologyIfAndOnlyIfItIsALinearContinuum
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更新时间:2025/5/4 8:23:27