# The Stacks Project

## Tag 01J9

Lemma 25.13.3. Let $X$ be a scheme. Points of $X$ correspond bijectively to equivalence classes of morphisms from spectra of fields into $X$. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element $\mathop{\rm Spec}(\kappa(x)) \to X$.

Proof. Follows from the discussion above. $\square$

The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 2308–2315 (see updates for more information).

\begin{lemma}
\label{lemma-characterize-points}
Let $X$ be a scheme. Points of $X$ correspond bijectively
to equivalence classes of morphisms from spectra of
fields into $X$. Moreover, each equivalence class contains
a (unique up to unique isomorphism) smallest element
$\Spec(\kappa(x)) \to X$.
\end{lemma}

\begin{proof}
Follows from the discussion above.
\end{proof}

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