ag.algebraic geometry - When is a morphism proper?

Assume $V$ and $W$ are quasiprojective. Let $i:Vto X$ be a locally closed embedding with $X$ projective (for instance $X$ could be $P^n$). Consider the induced map $g:Vto Xtimes W$; this is also a locally closed embedding. Then $f$ is proper iff $g$ is a closed embedding, or equivalently if $g(V)$ is closed.

As for the topological approach, use the definition of properness given by Charles Staats. Let $f:Xto Y$ be a continuous map of Hausdorff second countable topological spaces. The base change $f':X'to Y'$ of $f$ by a continuous map $g:Y'to Y$ is defined by letting $X'$ be the set of pairs
$(x,y')$ in $Xtimes Y'$ such that $f(x)=g(y')$ (with the induced topology from $Xtimes Y'$), and $f':X'to Y'$ the obvious projection. Then $f$ is proper if and only if all its base changes are closed. This may not be logically relevant, but I find it very comforting.
To connect the two cases note that, given a locally closed embedding of complex algebraic varieties, it is closed in the Zariski topology iff it is closed in the Euclidean topology.

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