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 \(p(x,y)=x^2=y^2-1\)then \(Z(p)\) being the unit circle is compact. If \(p(x,y)=x\) then \(Z(p)\) is the Y -axis, which is certainly non-compact.