[proofplan] We need a single fact: in a topological space, the preimage of a closed set under a continuous map is closed, and finite intersections of closed sets are closed. The Segre embedding identifies $\mathbb{P}^n_k \times \mathbb{P}^m_k$ with the Segre variety $S_{n,m}$, which is itself closed in $\mathbb{P}^N_k$ by [Image of the Segre Embedding](/theorems/2145). The projections $\pi_1, \pi_2$ are morphisms hence continuous in the Zariski topology. Therefore $\pi_1^{-1}(V) \subset S_{n,m}$ and $\pi_2^{-1}(W) \subset S_{n,m}$ are closed in $S_{n,m}$, their intersection $V \times W$ is closed in $S_{n,m}$, and since $S_{n,m}$ is closed in $\mathbb{P}^N_k$, $V \times W$ is closed in $\mathbb{P}^N_k$ as well. [/proofplan]

[step:Use the Segre embedding to view $\mathbb{P}^n_k \times \mathbb{P}^m_k$ as a closed subset of $\mathbb{P}^N_k$] By [Image of the Segre Embedding](/theorems/2145), the Segre map \begin{align*} \Sigma_{n,m} : \mathbb{P}^n_k \times \mathbb{P}^m_k \to \mathbb{P}^N_k \end{align*} is injective and its image is the Zariski-closed subvariety \begin{align*} S_{n,m} = V(\{Z_{ij} Z_{kl} - Z_{il} Z_{kj} : 0 \le i, k \le n,\, 0 \le j, l \le m\}) \subset \mathbb{P}^N_k, \end{align*} the locus of $2 \times 2$-minor relations on the $(n+1) \times (m+1)$ matrix of homogeneous coordinates $(Z_{ij})$. Via $\Sigma_{n,m}$ we transfer the Zariski topology of $S_{n,m}$ (induced as a subspace of $\mathbb{P}^N_k$) to $\mathbb{P}^n_k \times \mathbb{P}^m_k$, and from this point on we identify \begin{align*} \mathbb{P}^n_k \times \mathbb{P}^m_k \cong S_{n,m} \subset \mathbb{P}^N_k. \end{align*} The product $V \times W \subset \mathbb{P}^n_k \times \mathbb{P}^m_k$ corresponds under this identification to a subset of $S_{n,m}$, and to prove it is a projective variety it suffices to show it is closed in $\mathbb{P}^N_k$. [/step]

[step:Identify the projections $\pi_1, \pi_2$ as morphisms on the Segre variety] The projections $\pi_1 : \mathbb{P}^n_k \times \mathbb{P}^m_k \to \mathbb{P}^n_k$ and $\pi_2 : \mathbb{P}^n_k \times \mathbb{P}^m_k \to \mathbb{P}^m_k$ are defined on $S_{n,m}$ as follows. For a point $[Z_{ij}] \in S_{n,m}$ corresponding to $([x], [y])$ with $Z_{ij} = x_i y_j$, the projections recover $x$ and $y$ from the matrix $[Z_{ij}]$. \textbf{Description of $\pi_1$.} On the open set $U_j = \{Z_{0j} \neq 0\} \cap S_{n,m}$ — which corresponds to points where $y_j \neq 0$ — the first projection is given by reading off the $j$-th column of the matrix: \begin{align*} \pi_1|_{U_j} : U_j &\to \mathbb{P}^n_k, \\ [Z_{ij}] &\mapsto [Z_{0j} : Z_{1j} : \cdots : Z_{nj}]. \end{align*} This is well-defined on $U_j$: the tuple $(Z_{0j}, \ldots, Z_{nj})$ is nonzero because $Z_{0j} \neq 0$; and it is the projective class of the $j$-th column, which equals $y_j x$ for the unique factorisation $Z_{ij} = x_i y_j$ from [Image of the Segre Embedding](/theorems/2145), so $[Z_{0j} : \cdots : Z_{nj}] = [x]$ since the global scaling by $y_j \in k^\times$ does not change the projective class. The formulas on overlapping charts $U_j \cap U_{j'}$ agree projectively because the $j$-th and $j'$-th columns are both nonzero scalar multiples of $x$ on the overlap, by the same factorisation. The open sets $\{U_j : 0 \le j \le m\}$ cover $S_{n,m}$ because every point $[x] \times [y]$ has at least one $y_j \neq 0$, equivalently at least one $Z_{0j} \neq 0$ (using $Z_{0j} = x_0 y_j$ and choosing the chart with $x_0 \neq 0$ also; if $x_0 = 0$, choose another nonzero $x_i$ and use the chart $\{Z_{ij} \neq 0\}$ for the appropriate $i, j$). The slight modification needed here — using charts of the form $\{Z_{ij_0} \neq 0\}$ for fixed $j_0$ but varying $i$ — gives a cover, and the projections are defined identically as "the column of $i$" in each chart. The map $\pi_1$ is therefore a morphism of projective varieties (it is given by homogeneous polynomials of degree $1$ on each chart of an open cover, with consistent projective values on overlaps), and in particular continuous in the Zariski topology. \textbf{Description of $\pi_2$.} Symmetrically, on $\{Z_{i0} \neq 0\} \cap S_{n,m}$, \begin{align*} \pi_2 : S_{n,m} &\to \mathbb{P}^m_k, \\ [Z_{ij}] &\mapsto [Z_{i0} : Z_{i1} : \cdots : Z_{im}], \end{align*} the $i$-th row of $[Z_{ij}]$. The same reasoning shows this is a well-defined morphism (continuous map) of projective varieties. [/step]

[step:Take preimages under continuous maps to obtain closed sets]Since $V \subset \mathbb{P}^n_k$ is closed and $\pi_1 : S_{n,m} \to \mathbb{P}^n_k$ is continuous (Step 2), the preimage \begin{align*} \pi_1^{-1}(V) \subset S_{n,m} \end{align*} is closed in $S_{n,m}$. Symmetrically, since $W \subset \mathbb{P}^m_k$ is closed and $\pi_2$ is continuous, \begin{align*} \pi_2^{-1}(W) \subset S_{n,m} \end{align*} is closed in $S_{n,m}$.[/step]

[guided]We are about to use only the formal property "preimage of closed under continuous is closed". Both projections $\pi_1, \pi_2$ have been verified to be morphisms in Step 2, hence continuous in the Zariski topology — this is a basic fact about morphisms of varieties: pullback of a homogeneous polynomial along a morphism is a polynomial, so the preimage of $V(F) = \{F = 0\}$ is $\{F \circ \pi = 0\} = V(F \circ \pi)$, again a closed set. We can also describe the closed sets $\pi_1^{-1}(V)$ and $\pi_2^{-1}(W)$ very concretely in $\mathbb{P}^N_k$. Choose homogeneous polynomials $f_1, \ldots, f_a \in k[X_0, \ldots, X_n]$ generating $I(V) \subset k[X_0, \ldots, X_n]$. On each chart $\{Z_{0j} \neq 0\} \cap S_{n,m}$, the projection $\pi_1$ sends $[Z]$ to $[Z_{0j} : \cdots : Z_{nj}]$, so the preimage of $V(f_\alpha)$ is the locus where $f_\alpha(Z_{0j}, \ldots, Z_{nj}) = 0$. The polynomial $\tilde{f}_\alpha^j(Z) := f_\alpha(Z_{0j}, Z_{1j}, \ldots, Z_{nj})$ is a homogeneous polynomial of degree $\deg f_\alpha$ in the $Z_{ij}$. Hence on this chart, \begin{align*} \pi_1^{-1}(V) \cap \{Z_{0j} \neq 0\} \cap S_{n,m} = V(\tilde{f}_1^j, \ldots, \tilde{f}_a^j) \cap \{Z_{0j} \neq 0\} \cap S_{n,m}, \end{align*} a closed set in this open chart. Glueing across the cover (and using the analogous formulas for the other charts) shows $\pi_1^{-1}(V)$ is closed in $S_{n,m}$. The geometric content is clean: an $S_{n,m}$-point $[Z]$ has $\pi_1([Z]) \in V$ iff one (equivalently, every nonzero) column of the matrix $[Z_{ij}]$ lies in $V$ as a point of $\mathbb{P}^n_k$ — and "every nonzero column lies in $V$" is a Zariski-closed condition because being in $V$ is closed.[/guided]

[step:Take intersections to obtain $V \times W$ and conclude it is closed in $\mathbb{P}^N_k$] The intersection of two closed subsets of a topological space is closed. Hence \begin{align*} V \times W = \pi_1^{-1}(V) \cap \pi_2^{-1}(W) \subset S_{n,m} \end{align*} is closed in $S_{n,m}$. By Step 1, $S_{n,m}$ itself is closed in $\mathbb{P}^N_k$. Closedness is transitive in subspace topologies: a set $A \subset S_{n,m}$ closed in the subspace topology means $A = S_{n,m} \cap C$ for some $C \subset \mathbb{P}^N_k$ closed; then $A$ is closed in $\mathbb{P}^N_k$ as the intersection of two closed sets $S_{n,m}$ and $C$. Applying this with $A = V \times W$: \begin{align*} V \times W \text{ closed in } \mathbb{P}^N_k. \end{align*} It remains to verify that $V \times W$ as a subset of $S_{n,m}$ corresponds, under the identification $S_{n,m} \cong \mathbb{P}^n_k \times \mathbb{P}^m_k$, to the set-theoretic Cartesian product $\{(p, q) : p \in V, q \in W\}$. This is immediate from the definitions of $\pi_1$ and $\pi_2$: under the identification, $\pi_1$ sends $(p, q) \mapsto p$ and $\pi_2$ sends $(p, q) \mapsto q$, so $\pi_1^{-1}(V) = \{(p, q) : p \in V\} = V \times \mathbb{P}^m_k$ and $\pi_2^{-1}(W) = \mathbb{P}^n_k \times W$, with intersection $V \times W$. Hence $V \times W \subset \mathbb{P}^N_k$ is a Zariski-closed subset of projective space, i.e. a projective variety, completing the proof. [/step]