Let p be a polynomial in the real variables x,y. Then the zero set Z(p) of p is always closed. However, Z(p) may or may not be compact. For instance, if p(x,y)=x2=y2−1then Z(p) being the unit circle is compact. If p(x,y)=x then Z(p) is the Y -axis, which is certainly non-compact.
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