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In $\mathbb{C}\mathbb{P}^2$ we define coordinate triangle to be the one with sides $\{x_0=0\}, \{x_1=0\}$ and $\{x_2=0\}$. How would you define its interior? What kind of equation should it satisfy?

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    Once you remove one of the lines, you are left with the usual affine plane $\mathbb C^2$; if you next remove the other two, which are the coordinate axes, what you get is connected. In other words, the complement of the three lines $\{x_j=0\}$, $j=0,1,2$, is connected, and it does not have any sensible *interior* to speak of.2019-04-16

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“Being interior” corresponds to some form of “ordering comparison”. Most usually you have some equation which is zero at the boundary, and you define that the interior is where the equation is positive. But you don't have a sign or an ordering for complex numbers. Therefore you cannot reasonably define interior.

You can use some function to turn complex numbers into real ones, e.g. by simply taking the real part only. But that will result in points being exterior and other points being interior with no boundary point in between these two. Except if you define boundary using this other function as well, in which case you'll no longer have the triangle boundary equations you stated in your question.

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    @MarianoSuárez-Alvarez, do you mean a Jordan Curve in $\mathbb C^1$? One example would be the unit circle, which has the equation $1 - \lvert z\rvert = 0$. It has a clear interior because its left hand side is a real-valued function, as opposed to the complex-valued left hand side in the $x_j=0$ equation from the question statement. One could change the question statement to $\lvert x_j\rvert=0$ but this would make all points “outside” except for those incicent with one of the lines. I fear I don't fully understand your objection yet.2012-09-28