. . "Sorgenfrey plane"@en . "1084951858"^^ . "In topologia, il piano di Sorgenfrey \u00E8 un controesempio spesso citato per confutare congetture apparentemente plausibili. Consiste nel prodotto della retta di Sorgenfrey (la retta reale dotata della topologia del limite inferiore) con se stessa. La retta e il piano di Sorgenfrey prendono il nome dal matematico statunitense ."@it . . . "En el \u00E1mbito de la topolog\u00EDa, el plano de Sorgenfrey a menudo es mencionado como un contraejemplo de muchas conjeturas que parecer\u00EDan plausibles. El mismo consiste del de dos copias de la , que es la bajo el intervalo topol\u00F3gico semiabierto. La l\u00EDnea y el plano de Sorgenfrey han sido nombrados en honor al matem\u00E1tico estadounidense Robert Sorgenfrey."@es . "Sorgenfrey-Ebene"@de . . "3105"^^ . "In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey."@en . . . . . . . "Die Sorgenfrey-Ebene ist ein nach dem Mathematiker Robert Henry Sorgenfrey benanntes Beispiel aus dem mathematischen Teilgebiet der Topologie."@de . . . "Plano de Sorgenfrey"@pt . . . "\u5728\u62D3\u64B2\u5B78\uFF0CSorgenfrey\u5E73\u9762\u662F\u4E00\u4E2A\u7D93\u5E38\u5F15\u7528\u5230\u7684\u53CD\u4F8B\u3002\u5B83\u662F\u5169\u689DSorgenfrey\u7DDA\u7684\u7A4D\uFF08Sorgenfrey\u7DDA\u662F\u8CE6\u4E88\u4E86\u4E0B\u9650\u62D3\u64B2\u7684\u5BE6\u6578\u7DDA\uFF09\u3002Sorgenfrey\u7DDA\u548CSorgenfrey\u5E73\u9762\u662F\u4EE5\u7F8E\u570B\u6578\u5B78\u5BB6 Robert Sorgenfrey\u547D\u540D\u3002 Sorgenfrey\u5E73\u9762\uFF08\u73FE\u5728\u958B\u59CB\u7528 \u8868\u793A\uFF09\u7684\u5176\u4E2D\u4E00\u7D44\u57FA\u662F\u6240\u6709\u300C\u5305\u542B\u5DE6\u908A\u3001\u5DE6\u4E0B\u9802\u9EDE\u3001\u4E0B\u908A\u800C\u4E0D\u5305\u542B\u5176\u4ED6\u908A\u3001\u9802\u9EDE\u300D\u7684\u9577\u65B9\u5F62\u3002\u4E0A\u7684\u958B\u96C6\u5247\u662F\u9019\u7A2E\u9577\u65B9\u5F62\u7684\u4E26\u96C6\u3002 \u80FD\u4F5C\u70BA\u5F88\u591A\u62D3\u64B2\u5B78\u4E0A\u807D\u8D77\u4F86\u5F88\u53EF\u80FD\u6B63\u78BA\u7684\u9673\u8FF0\u7684\u53CD\u4F8B\u5B50\u3002\u5176\u4E00\uFF0C\u5B83\u662F\u6797\u5FB7\u52D2\u592B\u7A7A\u9593\u7684\u7A4D\uFF0C\u4F46\u5B83\u81EA\u5DF1\u4E0D\u662F\u6797\u5FB7\u52D2\u592B\u7A7A\u9593\u3002\u5176\u4E8C\uFF0C\u53CD\u5C0D\u89D2\u7DDA\u662F\u4E0A\u7684\u4E00\u500B\u4E0D\u53EF\u6578\u96E2\u6563\u5B50\u96C6\uFF0C\u6240\u4EE5\u5B83\u662F\u4E0D\u53EF\u5206\u7684\uFF0C\u4F46\u662F\u53EF\u5206\u7684\u3002\u9019\u500B\u4F8B\u5B50\u5C55\u793A\u4E86\u53EF\u5206\u7A7A\u9593\u7684\u9589\u5B50\u96C6\u4E0D\u4E00\u5B9A\u662F\u53EF\u5206\u7684\u3002\u5176\u4E09\uFF0C\u548C\u662F\u9589\u96C6\uFF0C\u800C\u4E14\u53EF\u4EE5\u8B49\u660E\u5B83\u5011\u4E0D\u80FD\u88AB\u9130\u57DF\u5206\u96E2\uFF0C\u6240\u4EE5\u4E0D\u662F\u6B63\u5247\u7A7A\u9593\u3002\u9019\u5C55\u793A\u4E86\u6B63\u5247\u7A7A\u9593\u7684\u7A4D\u4E0D\u4E00\u5B9A\u662F\u6B63\u5247\u7684\uFF0C\u751A\u81F3\u5C55\u793A\u4E86\u66F4\u5F37\u7684\u7D50\u679C\uFF1A\u6709\u9650\u500B\u5B8C\u7F8E\u6B63\u5247\u7A7A\u9593\u7684\u7A4D\u4E5F\u4E0D\u4E00\u5B9A\u662F\u6B63\u5247\u7684\u3002"@zh . . . . . . . . . "682853"^^ . . . . "In topologia, il piano di Sorgenfrey \u00E8 un controesempio spesso citato per confutare congetture apparentemente plausibili. Consiste nel prodotto della retta di Sorgenfrey (la retta reale dotata della topologia del limite inferiore) con se stessa. La retta e il piano di Sorgenfrey prendono il nome dal matematico statunitense . Una base per il piano di Sorgenfrey, denotato d'ora in poi con , \u00E8 costituita dall'insieme dei rettangoli che includono il lato sinistro, lo spigolo sinistro inferiore e il lato inferiore mentre non includono lo spigolo inferiore destro, il lato destro, lo spigolo superiore destro, il lato superiore e lo spigolo superiore sinistro. Gli aperti di questa topologia sono costituiti dalle unioni di tali rettangoli. \u00E8 un esempio di spazio non di Lindel\u00F6f ma che \u00E8 prodotto di spazi di Lindel\u00F6f. \u00C8 anche un esempio di spazio non normale ma che \u00E8 prodotto di spazi normali. Di questo spazio consideriamo la diagonale secondaria , questo \u00E8 un sottoinsieme discreto che come sottospazio topologico risulta non essere separabile nonostante il piano di Sorgenfrey lo sia. Ci\u00F2 dimostra che la separabilit\u00E0 non \u00E8 ereditata dalla topologia del sottoinsieme. Da notare che e sono insiemi chiusi che non possono essere separati con insiemi aperti; ci\u00F2 mostra che non \u00E8 uno spazio normale."@it . "Plano de Sorgenfrey"@es . "Em topologia, o plano de Sorgenfrey \u00E9 frequentemente citado como contra-exemplo para v\u00E1rias conjecturas de outro modo plaus\u00EDveis. Ele consiste do espa\u00E7o produto de duas c\u00F3pias da linha de Sorgenfrey, que \u00E9 a reta real com a topologia dos intervalos semi-abertos. A reta de Sorgenfrey e seu plano recebem tal nome em homenagem ao matem\u00E1tico Americano Robert Sorgenfrey."@pt . . . . . . "En math\u00E9matiques, le plan de Sorgenfrey est un espace topologique souvent utilis\u00E9, \u00E0 plusieurs titres, comme contre-exemple. C'est le produit S\u00D7S de la droite de Sorgenfrey S par elle-m\u00EAme. Robert Sorgenfrey a d\u00E9montr\u00E9 que le plan S\u00D7S est non normal (donc non paracompact), tandis que la droite S est paracompacte (donc normale)."@fr . . . . . . . . . . "Em topologia, o plano de Sorgenfrey \u00E9 frequentemente citado como contra-exemplo para v\u00E1rias conjecturas de outro modo plaus\u00EDveis. Ele consiste do espa\u00E7o produto de duas c\u00F3pias da linha de Sorgenfrey, que \u00E9 a reta real com a topologia dos intervalos semi-abertos. A reta de Sorgenfrey e seu plano recebem tal nome em homenagem ao matem\u00E1tico Americano Robert Sorgenfrey. Uma base para o plano de Sorgenfrey, denotado por a partir de agora, \u00E9 portanto o conjunto dos ret\u00E2ngulos que incluem o lado esquerdo, o canto sudoeste, e o lado inferior, e omite o canto sudeste, o lado direito, o canto nordeste, o lado superior, e o canto noroeste. Abertos de s\u00E3o uni\u00F5es de tais ret\u00E2ngulos. \u00E9 um exemplo de um espa\u00E7o que \u00E9 produto de espa\u00E7os de Lindel\u00F6f que n\u00E3o \u00E9 um espa\u00E7o de Lindel\u00F6f. \u00C9 tamb\u00E9m um exemplo de um espa\u00E7o que \u00E9 produto de espa\u00E7os normais e n\u00E3o \u00E9 normal.A chamada , \u0394 = { (x, \u2212x) | x \u2208 R } \u00E9 um deste espa\u00E7o, e \u00E9 um do espa\u00E7o separ\u00E1vel X. Isto mostra que a separabilidade n\u00E3o \u00E9 herdada para subespa\u00E7os fechados. Note que K = { (x, \u2212x) | x \u2208 Q } e\u0394\\K s\u00E3o conjuntos fechados que n\u00E3o podem ser separados por conjuntos abertos, mostrando que X n\u00E3o \u00E9 normal."@pt . "En math\u00E9matiques, le plan de Sorgenfrey est un espace topologique souvent utilis\u00E9, \u00E0 plusieurs titres, comme contre-exemple. C'est le produit S\u00D7S de la droite de Sorgenfrey S par elle-m\u00EAme. Robert Sorgenfrey a d\u00E9montr\u00E9 que le plan S\u00D7S est non normal (donc non paracompact), tandis que la droite S est paracompacte (donc normale)."@fr . "\u5728\u62D3\u64B2\u5B78\uFF0CSorgenfrey\u5E73\u9762\u662F\u4E00\u4E2A\u7D93\u5E38\u5F15\u7528\u5230\u7684\u53CD\u4F8B\u3002\u5B83\u662F\u5169\u689DSorgenfrey\u7DDA\u7684\u7A4D\uFF08Sorgenfrey\u7DDA\u662F\u8CE6\u4E88\u4E86\u4E0B\u9650\u62D3\u64B2\u7684\u5BE6\u6578\u7DDA\uFF09\u3002Sorgenfrey\u7DDA\u548CSorgenfrey\u5E73\u9762\u662F\u4EE5\u7F8E\u570B\u6578\u5B78\u5BB6 Robert Sorgenfrey\u547D\u540D\u3002 Sorgenfrey\u5E73\u9762\uFF08\u73FE\u5728\u958B\u59CB\u7528 \u8868\u793A\uFF09\u7684\u5176\u4E2D\u4E00\u7D44\u57FA\u662F\u6240\u6709\u300C\u5305\u542B\u5DE6\u908A\u3001\u5DE6\u4E0B\u9802\u9EDE\u3001\u4E0B\u908A\u800C\u4E0D\u5305\u542B\u5176\u4ED6\u908A\u3001\u9802\u9EDE\u300D\u7684\u9577\u65B9\u5F62\u3002\u4E0A\u7684\u958B\u96C6\u5247\u662F\u9019\u7A2E\u9577\u65B9\u5F62\u7684\u4E26\u96C6\u3002 \u80FD\u4F5C\u70BA\u5F88\u591A\u62D3\u64B2\u5B78\u4E0A\u807D\u8D77\u4F86\u5F88\u53EF\u80FD\u6B63\u78BA\u7684\u9673\u8FF0\u7684\u53CD\u4F8B\u5B50\u3002\u5176\u4E00\uFF0C\u5B83\u662F\u6797\u5FB7\u52D2\u592B\u7A7A\u9593\u7684\u7A4D\uFF0C\u4F46\u5B83\u81EA\u5DF1\u4E0D\u662F\u6797\u5FB7\u52D2\u592B\u7A7A\u9593\u3002\u5176\u4E8C\uFF0C\u53CD\u5C0D\u89D2\u7DDA\u662F\u4E0A\u7684\u4E00\u500B\u4E0D\u53EF\u6578\u96E2\u6563\u5B50\u96C6\uFF0C\u6240\u4EE5\u5B83\u662F\u4E0D\u53EF\u5206\u7684\uFF0C\u4F46\u662F\u53EF\u5206\u7684\u3002\u9019\u500B\u4F8B\u5B50\u5C55\u793A\u4E86\u53EF\u5206\u7A7A\u9593\u7684\u9589\u5B50\u96C6\u4E0D\u4E00\u5B9A\u662F\u53EF\u5206\u7684\u3002\u5176\u4E09\uFF0C\u548C\u662F\u9589\u96C6\uFF0C\u800C\u4E14\u53EF\u4EE5\u8B49\u660E\u5B83\u5011\u4E0D\u80FD\u88AB\u9130\u57DF\u5206\u96E2\uFF0C\u6240\u4EE5\u4E0D\u662F\u6B63\u5247\u7A7A\u9593\u3002\u9019\u5C55\u793A\u4E86\u6B63\u5247\u7A7A\u9593\u7684\u7A4D\u4E0D\u4E00\u5B9A\u662F\u6B63\u5247\u7684\uFF0C\u751A\u81F3\u5C55\u793A\u4E86\u66F4\u5F37\u7684\u7D50\u679C\uFF1A\u6709\u9650\u500B\u5B8C\u7F8E\u6B63\u5247\u7A7A\u9593\u7684\u7A4D\u4E5F\u4E0D\u4E00\u5B9A\u662F\u6B63\u5247\u7684\u3002"@zh . "Die Sorgenfrey-Ebene ist ein nach dem Mathematiker Robert Henry Sorgenfrey benanntes Beispiel aus dem mathematischen Teilgebiet der Topologie."@de . . "In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line, which is the real line under the half-open interval topology. The Sorgenfrey line and plane are named for the American mathematician Robert Sorgenfrey. A basis for the Sorgenfrey plane, denoted from now on, is therefore the set of rectangles that include the west edge, southwest corner, and south edge, and omit the southeast corner, east edge, northeast corner, north edge, and northwest corner. Open sets in are unions of such rectangles. is an example of a space that is a product of Lindel\u00F6f spaces that is not itself a Lindel\u00F6f space. The so-called anti-diagonal is an uncountable discrete subset of this space, and this is a non-separable subset of the separable space . It shows that separability does not inherit to closed subspaces. Note that and are closed sets; it can be proved that they cannot be separated by open sets, showing that is not normal. Thus it serves as a counterexample to the notion that the product of normal spaces is normal; in fact, it shows that even the finite product of perfectly normal spaces need not be normal."@en . . . . . . "Piano di Sorgenfrey"@it . . . . . "Sorgenfrey\u5E73\u9762"@zh . . . "En el \u00E1mbito de la topolog\u00EDa, el plano de Sorgenfrey a menudo es mencionado como un contraejemplo de muchas conjeturas que parecer\u00EDan plausibles. El mismo consiste del de dos copias de la , que es la bajo el intervalo topol\u00F3gico semiabierto. La l\u00EDnea y el plano de Sorgenfrey han sido nombrados en honor al matem\u00E1tico estadounidense Robert Sorgenfrey. Una base del plano de Sorgenfrey, expresada como , es por lo tanto el grupo de rect\u00E1ngulos que incluyen el borde oeste, el v\u00E9rtice suroeste, y el borde sur, y omiten el v\u00E9rtice sureste, el borde este, el v\u00E9rtice noreste, el borde norte, y el v\u00E9rtice noroeste. Los conjuntos abiertos en son uniones de estos rect\u00E1ngulos."@es . "Plan de Sorgenfrey"@fr . . . .