Let $X \subset \mathbb{P}^n_{\mathbb{C}}$ be a [projective variety](/page/Projective%20Variety), meaning that there is a set $S \subset \mathbb{C}[x_0,\ldots,x_n]$ of homogeneous polynomials such that $X = V_+(S)$. Equip $\mathbb{P}^n_{\mathbb{C}}$ with its analytic topology and $X$ with the induced [subspace topology](/page/Subspace%20Topology). Then $X$ is compact.