| Information | |
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| has gloss | eng: The least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis stating that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. It is an axiom in the sense that it cannot be proven by the other axioms within the system of real analysis. This axiom is very useful since it is essential to the proof that the real number line is a complete metric space. The rational number line does not satisfy the LUB axiom and hence is not complete. |
| lexicalization | eng: Least upper bound axiom |
| instance of | (noun) (logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident axiom |
| Meaning | |
|---|---|
| Italian | |
| has gloss | ita: In matematica, lassioma di Dedekind, detto anche assioma di continuità oppure assioma di completezza, riguarda linsieme dei numeri reali R; esso afferma che ogni insieme S di numeri reali che non sia vuoto e che sia limitato superiormente possiede un estremo superiore, vale a dire un numero reale uguale o maggiore di tutti gli elementi di S e tale che non esista nessun reale più piccolo con tale proprietà. |
| lexicalization | ita: assioma di Dedekind |
| Korean | |
| has gloss | kor: 상한공리 또는 최소상계공리(least upper bound axiom)는 해석학의 공리중 하나로 자세한 내용은 다음과 같다. |
| lexicalization | kor: 최소상계공리 |
| Portuguese | |
| has gloss | por: O axioma do supremo é um axioma de continuidade. Ele é usado na construção analítica dos números reais. |
| lexicalization | por: Axioma do supremo |
| Chinese | |
| has gloss | zho: 最小上界公理，又稱為上確界原理，是实分析的公理。之所以稱為公理，是因為它在实分析的公理系统裡，不能被除了它本身以外的公理所證明。这个公理声称如果实数的非空子集有上界，则它有最小上界。这个公理可以用來证明实数集是完备度量空间。有理数集不满足最小上界公理，因而就不是完备的。一个理想的例子是 S = \ x\in \mathbbQ}|x^2 < 2\}。2 当然是这个集合的上界。但是这个集合没有最小上界 — 对于任何 x \in S，我们可以找到 y \in S 有着 y > x \ 。 |
| lexicalization | zho: 最小上界公理 |
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