Show that a morphism $ \varphi:X\rightarrow Y$ is a closed immersion iff $ \varphi(|X|)$ is closed in |Y|, $ \varphi^*:O_Y\rightarrow\varphi_*O_X$ is a surjection, $ |X|\rightarrow\varphi(|X|)$ is a homeomorphism and $ \ker\varphi^*$ is of finite type.

We use a quite uncommon definition of closed immersion: $ \varphi:X\rightarrow Y$ is a morphism of analytic spaces and there exists a closed analytic subspace Z of Y s.t. $ \varphi=\tau\circ\rho$ , where $ \tau:Z\rightarrow Y$ is the inclusion morphism and $ \rho:X\rightarrow Z$ is an isomorphism of analytic spaces.

I feel like “$ \rightarrow$ ” should be clear, but I don’t really know where to start and I apologize for already having posted this on MSE.