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单词 ClosedComplexPlane
释义

closed complex plane


The complex planeMathworldPlanetmath , i.e. the set of the complex numbersMathworldPlanetmathPlanetmath z satisfying

|z|<,

is open but not closed, since it doesn’t contain the accumulation pointsPlanetmathPlanetmath of all sets of complex numbers, for example of the set {1, 2, 3,}.  One can to the closed complex plane {} by adding to the infinite point which the lacking accumulation points. One settles that  ||=,  where the latter means the real infinity.

The resulting space is the one-point compactification of . The open sets are the open sets in together with sets containing whose complement is compact in . Conceptually, one thinks of the additional open sets as those open sets “around ”.

The one-point compactification of is also the complex projective line 1, as well as the Riemann sphere.

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更新时间:2025/5/4 16:17:49