Let $X$ be a scheme of finite type over $K$ and $X^{{\text{\rm an}}}$ the associated analytic space.

1. (1)

   $X^{{\text{\rm an}}}$ is a locally compact and locally arc-connected topological space.

2. (2)

   $X^{{\text{\rm an}}}$ is Hausdorff (respectively compact and Hausdorff, arc-connected) if and only if $X$ is separated (respectively proper, connected).

3. (3)

   The map $\pi\colon X^{{\text{\rm an}}}\to X$ is continuous. A locally constructible subset $T\subset X$ is open (respectively closed, dense) if and only if $\pi^{-1}(T)$ is open (respectively closed, dense).

4. (4)

   Let $\psi\colon X\longrightarrow Y$ be a morphism of schemes of finite type over $K$ and $\psi^{{\text{\rm an}}}\colon X^{{\text{\rm an}}}\longrightarrow Y^{{\text{\rm an}}}$ its analytification. Then $\psi$ is flat (respectively unramified, étale, smooth, separated, injective, surjective, open immersion, isomorphism) if and only if $\psi^{{\text{\rm an}}}$ has the same property.

5. (5)

   Let $K^{\prime}$ be a complete extension of $K$. Then the map $\pi_{K^{\prime}}:X^{{\text{\rm an}}}_{K^{\prime}}\to X_{K^{\prime}}$ induces a bijection between $X^{{\text{\rm an}}}(K^{\prime})$ and $X(K^{\prime})$.

6. (6)

   Set $X_{{\text{\rm alg}}}=\{p\in X|\,[K(p):K]<\infty\}$. Then $\pi$ induces a bijection between $X^{{\text{\rm an}}}_{{\text{\rm alg}}}$ and $X_{{\text{\rm alg}}}$. The subset $X^{{\text{\rm an}}}_{{\text{\rm alg}}}\subset X^{{\text{\rm an}}}$ is dense.