Section 0: Problem 2 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $f\in\mathbb{R}[X,Y]$ and let $C:(f=0)\subset\mathbb{R}^{2}$ ; say that $P\in C$ is isolated if there is an $\epsilon>0$ such that $C\cap B(P,\epsilon)=P$ . Show by example that $C$ can have isolated points. Prove that if $P\in C$ is an isolated point then $f:\mathbb{R}^{2}\to\mathbb{R}$ must have a max or min at $P$ and deduce that $\partial f/\partial x$ and $\partial f/\partial y$ vanish at $P$ . This proves that an isolated point of a real curve is singular.

For example, $f(x,y)=x^{2}+y^{2}$ has the only (isolated) point in $C$ . A more interesting example, $x^{3}-y^{2}-3x-2$ has one isolated point in $C$ , $(-1,0)$ .

Take a neighborhood in which there is one solution only, assume there are two points having a positive and negative values, draw a closed curve inside the neighborhood going through these points but not through the isolated point, and conclude using the mean value theorem for connected compact spaces that then there is another solution in the neighborhood. Therefore, since $f$ as a polynomial is differentiable the partial derivatives are zero.

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Section 0: Problem 2 Solution

Section 0: Problem 2 Solution