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'topology of the complex plane'
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| Title of object: |
topology of the complex plane |
| Canonical Name: |
TopologyOfTheComplexPlane |
| Type: |
Definition |
| Created on: |
2003-05-25 06:16:15 |
| Modified on: |
2009-04-28 21:20:13 |
| Classification: |
msc:30-00, msc:54E35 |
| Defines: |
open disk, accumulation point, interior point, open, closed |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}} |
Content:
The usual topology for the complex plane $\sC$
is the topology induced by the metric
$$d(x,\,y) := |x\!-\!y|$$
for\, $x,\,y \in \sC$.
Here, $|\cdot|$ is the \PMlinkname{complex modulus}{ModulusOfComplexNumber}.
If we identify $\sR^2$ and $\sC$, it is clear that the above
topology coincides with topology induced by the Euclidean metric on $\sR^2$.
Some basic topological concepts for $\sC$:
\begin{enumerate}
\item The open balls
$$B_r(\zeta) \;=\; \{z\in\sC\,\vdots\; |z\!-\!\zeta| < r\}$$
are often called \emph{open disks}.
\item A point $\zeta$ is an \emph{accumulation point} of a subset $A$ of $\sC$, if any open disk $B_r(\zeta)$ contains at least one point of $A$ distinct from $\zeta$.
\item A point $\zeta$ is an interior point of the set $A$, if there exists an open disk $B_r(\zeta)$ which is contained in $A$.
\item A set $A$ is \emph{open}, if each of its points is an interior point of $A$.
\item A set $A$ id \emph{closed}, if all its accumulation points belong to $A$.
\end{enumerate} |
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