... curve¹

Of course, the full designation is used only in formal situations. One normally adopts an abbreviated terminology. We shall say simply a curve in contexts where this will not lead to confusion.

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...crossing,²

Sometimes other names are used. For example: a node, a point of type $A_1$ with two real branches, a nonisolated nondegenerate double point.

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... pieces.³

Proof: the closure of tne union of a connected component of $\mathbb{C}A\smallsetminus \mathbb{R}A$ with its image under $conj$ is open and close in $\mathbb{C}A$, but $\mathbb{C}A$ is connected.

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... sphere.⁴

I believe that this may be assumed well-known. A short explanation is that a projective line is a one-point compactification of an affine line, which, in the complex case, is homeomorphic to $\mathbb{R}^2$. A one-point compactification of $\mathbb{R}^2$ is unique up to homeomorphism and homeomorphic to $S^2$.

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....⁵

Division by 2 appears here to make this notion closer to the well-known notion for an affine plane curve. In the definition for affine situation one uses a ray instead of entire line. In the projective situation there is no natural way to divide a line into two rays, but we still have an opportunity to divide the result by 2. Another distinction from the affine situation is that there the index may be negative. It is related to the fact that the affine plane is orientable, while the projective plane is not.

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... tree⁶

Here by the diameter of a tree it is understood the maximal number of edges in a simple chain of edges of the tree, i. e., the diameter of the tree in the internal metric, with respect to which each edge has length 1.

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