Connected subset of $\Bbb{R}^2$ and $\Bbb{C}^2$.

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Topologically there is no difference between $\mathbb{R}^2$ and $\mathbb{C}$.

The statement of your instructor is true for open connected subsets of $\mathbb{C}$ (so called open domains), not for all subsets.

The unit circle ( $\{z :|z|=1 \}$) is connected (we can find continuous paths between any two points, e.g. or we can write it as a continuous image of the connected space $\mathbb{R}$ using $t \to e^{it}$ etc.).

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Abhishek Chandra
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Abhishek Chandra

Persuing Undergraduate..lot to learn

Updated on August 01, 2022

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  • Abhishek Chandra
    Abhishek Chandra over 1 year

    During my complex analysis course. My instructor gave me the definition a set in Complex plane is said to be connected if any points of the set can be joined by polygon lines that should be contained in the set.

    I liked this definition very much. As it was helping a lot to check which sets are connected and which are not.

    I want to know is this definition also valid for real plane $\mathbb{R}^2$.

    Because today when I was doing a question to check whether a unit circle centered at origin is connected or not. I immediately said it is not connected as we cannot draw polygon lines between any two distinct points lying on the unit circle such that the polygon line belongs to the set. But the correct answer was connected.

    Please help me how is it possible. And I would also like to know is $|z|=1$ also connected (I am not taking the part inside circle) just the boundary.

    Please help me clear my doubt.