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Hometopology of the complex plane

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# topology of the complex plane

The usual topology for the complex plane $\mathbb{C}$ is the topology induced by the metric

$d(x,\,y):=|x\!-\!y|$ |

for $x,\,y\in\mathbb{C}$. Here, $|\cdot|$ is the complex modulus.

If we identify $\mathbb{R}^{2}$ and $\mathbb{C}$, it is clear that the above topology coincides with topology induced by the Euclidean metric on $\mathbb{R}^{2}$.

Some basic topological concepts for $\mathbb{C}$:

1. The open balls

$B_{r}(\zeta)\;=\;\{z\in\mathbb{C}\,\vdots\;|z\!-\!\zeta|<r\}$ are often called

*open disks*.2. A point $\zeta$ is an

*accumulation point*of a subset $A$ of $\mathbb{C}$, if any open disk $B_{r}(\zeta)$ contains at least one point of $A$ distinct from $\zeta$.3. A point $\zeta$ is an

*interior point*of the set $A$, if there exists an open disk $B_{r}(\zeta)$ which is contained in $A$.4. A set $A$ is

*open*, if each of its points is an interior point of $A$.5. A set $A$ is

*closed*, if all its accumulation points belong to $A$.6. A set $A$ is

*bounded*, if there is an open disk $B_{r}(\zeta)$ containing $A$.7. A set $A$ is

*compact*, if it is closed and bounded.

## Mathematics Subject Classification

54E35*no label found*30-00

*no label found*

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