Sorgenfrey half-open plane

The Sorgenfrey plane is the product of the Sorgenfrey line with itself. This topology can also be described as the topology on $\\mathbb{R}^{2}$ which arises from the basis $\\{[a,b)\\times[c,d)\\mid a,b,c,d\\in\\mathbb{R},a<b,c<d\\}$ .

It is interesting to note that, even though the Sorgenfrey lineenjoys the Lindelöf property (http://planetmath.org/lindelofspace), theSorgenfrey plane does not. To see this, one can note that theline $x+y=0$ is a closed subset in this topology and thatthe induced topology on this line is the discrete topology.Since the Lindelöf property is weakly hereditary, thediscrete topology on the real line would have to be Lindelöfif the Sorgenfrey plane topology were Lindelöf. However, thediscrete topology on an uncountable set can never have theLindelöf property, so the Sorgenfrey topology cannot havethis property either.

Reference

Sorgenfrey, R. H. On the Topological Product of Paracompact Spaces, Bulletin of the American Mathematical Society, (1947) 631-632

单词	SorgenfreyHalfopenPlane
释义	Sorgenfrey half-open plane The Sorgenfrey plane is the product of the Sorgenfrey line with itself. This topology can also be described as the topology on $\\mathbb{R}^{2}$ which arises from the basis $\\{[a,b)\\times[c,d)\\mid a,b,c,d\\in\\mathbb{R},a<b,c<d\\}$ . It is interesting to note that, even though the Sorgenfrey lineenjoys the Lindelöf property (http://planetmath.org/lindelofspace), theSorgenfrey plane does not. To see this, one can note that theline $x+y=0$ is a closed subset in this topology and thatthe induced topology on this line is the discrete topology.Since the Lindelöf property is weakly hereditary, thediscrete topology on the real line would have to be Lindelöfif the Sorgenfrey plane topology were Lindelöf. However, thediscrete topology on an uncountable set can never have theLindelöf property, so the Sorgenfrey topology cannot havethis property either. Reference Sorgenfrey, R. H. On the Topological Product of Paracompact Spaces, Bulletin of the American Mathematical Society, (1947) 631-632
随便看	negative definite NegativeHypergeometricRandomVariable negative hypergeometric random variable NegativeHypergeometricRandomVariableExampleOf negative hypergeometric random variable, example of NegativeNumber negative number NegativeSemidefinite negative semidefinite NegativeTranslation negative translation Neighborhood neighborhood NeighborhoodofAVertex neighborhood (of a vertex) NeighborhoodRetract neighborhood retract NeighborhoodSystem neighborhood system NeighborhoodSystemOnASet neighborhood system on a set NeilJASloane Neil J. A. Sloane NerodeEquivalence Nerode equivalence