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单词 TopologyOfTheComplexPlane
释义

topology of the complex plane


The usual topology for the complex planeMathworldPlanetmath is the topologyMathworldPlanetmath induced by the metric

d(x,y):=|x-y|

for  x,y.Here, || is the complex modulusMathworldPlanetmath (http://planetmath.org/ModulusOfComplexNumber).

If we identify 2 and , it is clear that the abovetopology coincides with topology induced by the Euclidean metric on 2.

Some basic topological concepts for :

  1. 1.

    The open ballsPlanetmathPlanetmath

    Br(ζ)={z|z-ζ|<r}

    are often called open disks.

  2. 2.

    A point ζ is an accumulation point of a subset A of , if any open disk Br(ζ) contains at least one point of A distinct from ζ.

  3. 3.

    A point ζ is an interior point of the set A, if there exists an open disk Br(ζ) which is contained in A.

  4. 4.

    A set A is open, if each of its points is an interior point of A.

  5. 5.

    A set A is closed, if all its accumulation points belong to A.

  6. 6.

    A set A is boundedPlanetmathPlanetmathPlanetmath, if there is an open disk Br(ζ) containing A.

  7. 7.

    A set A is compact, if it is closed and bounded.

Titletopology of the complex plane
Canonical nameTopologyOfTheComplexPlane
Date of creation2013-03-22 13:38:40
Last modified on2013-03-22 13:38:40
Ownermatte (1858)
Last modified bymatte (1858)
Numerical id8
Authormatte (1858)
Entry typeDefinition
Classificationmsc 54E35
Classificationmsc 30-00
Related topicIdentityTheorem
Related topicPlacesOfHolomorphicFunction
Definesopen disk
Definesaccumulation point
Definesinterior point
Definesopen
Definesclosed
Definesbounded
Definescompact
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更新时间:2025/5/4 22:13:27