TITLE: Topological space underlying this curve
QUESTION [0 upvotes]: I have to solve this exercise but I have really no clue even how to start with it:
Identify the topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$. How does it fit with the shape of $Y^2Z=X(X-Z)(X-\lambda Z)$ in $\mathbb{R}^2$ ($\lambda\in\mathbb{R}-\{0,1\}$)?
I would be really thankful if anybody could help me, thanks in advance!

REPLY [1 votes]: a) The  topological space underlying the cubic $Y^2Z=X^2(X-Z)$ in $\mathbb{PR}^2$ is disconnected .
One (closed) connected component is the singleton set consisting of just the origin {[0:0:1]}.
The other  component (the rest) is homeomorphic to a circle.  
b) Every  curve $Y^2Z=X(X-Z)(X-\lambda Z)$ with $\mathbb{R}^2$ ($\lambda\in\mathbb{R}-\{0,1\}$) is disconnected too, and is homeomorphic to the disjoint union of two circles.
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