Let $X \subset \mathbb A^n$, $W \subset \Bbb A^m$ be two algebraic sets. A function $\phi:X \rightarrow W$ is a morphism if there exist $m$ polynomial functions $f_1,\ldots,f_m \in K[X]$ such that for every $x \in X$ one has $ \phi(x)=(f_1(x),\ldots,f_m(x)) \in W$.
We say two algebraic sets are isomorphic if there exists a bijective morphism between them whose inverse is also a morphism. Now, now my question is what does ismorphism of two algebraic sets mean intuitively? The general meaning of isomorphism is that they are "equivalent".So, in what geometrical sense two isomorphic algebraic sets can be considered "equivalent"? In other words.what geometrical properties are same for two isomorphic algebraic sets? Also, can a Zariski open set in one Zariski topology be isomorphic to a Zariski closed in another Zariski topology?