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If $X$ is a scheme and $x \in X$, then the canonical morphism $j_x : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is a homeomorphism onto its image. The image consists of those $y \in X$ which generalize $x$, i.e. satisfy $x \in \overline{\{y\}}$. Locally, this is just the classification of prime ideals in the localization.

It follows that that the image is closed under generalizations, and contains $x$. In general, I doubt that anything more can be said.

Every open subset is stable under generalizations. The converse is true for constructible subsets of noetherian sober topological spaces (see here). Chevalley's theorem implies that the image of a morphism of finite presentation is constructible. However, $j_x$ is almost never of finite presentation.

If $X$ is a scheme and $x \in X$, then the canonical morphism $j_x : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is a homeomorphism onto its image. The image consists of those $y \in X$ which generalize $x$, i.e. satisfy $x \in \overline{\{y\}}$. Locally, this is just the classification of prime ideals in the localization.

It follows that that the image is closed under generalizations, and contains $x$. In general, I doubt that anything more can be said.

Every open subset is stable under generalizations. The converse is true for constructible subsets of noetherian sober topological spaces (see here). Chevalley's theorem implies that the image of a morphism of finite presentation is constructible. However, $j_x$ is almost never of finite presentation.

For example, when $x$ is a generic point of $X$, this means that the image of $j_x$ equals $\{x\}$, which is usually neither open nor closed.

If $X$ is a scheme and $x \in X$, then the canonical morphism $j_x : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is a homeomorphism onto its image. The image consists of those $y \in X$ which generalize $x$, i.e. satisfy $x \in \overline{\{y\}}$. Locally, this is just the classification of prime ideals in the localization.

It follows that that the image is closed under generalizations, and contains $x$. In general, I doubt that anything more can be said.

Every open subset is stable under generalizations. The converse is true for constructible subsets of noetherian sober topological spaces (see here). Chevalley's theorem implies that the image of a morphism of finite presentation is constructible. In particular, when $x$ is a closed point and $X$ is noetherian, the image of $j_x$ is open.

If $X$ is a scheme and $x \in X$, then the canonical morphism $j_x : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is a homeomorphism onto its image. The image consists of those $y \in X$ which generalize $x$, i.e. satisfy $x \in \overline{\{y\}}$. Locally, this is just the classification of prime ideals in the localization.

It follows that that the image is closed under generalizations, and contains $x$. In general, I doubt that anything more can be said.

Every open subset is stable under generalizations. The converse is true for constructible subsets of noetherian sober topological spaces (see here). Chevalley's theorem implies that the image of a morphism of finite presentation is constructible. However, $j_x$ is almost never of finite presentation.

If $X$ is a scheme and $x \in X$, then the canonical morphism $\mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is a homeomorphism onto its image. The image consists of those $y \in X$ which generalize $x$, i.e. satisfy $x \in \overline{\{y\}}$. Locally, this is just the classification of prime ideals in the localization.

It follows that that the image is closed under generalizations, and contains $x$. I doubt that anything more can be said.

If $X$ is a scheme and $x \in X$, then the canonical morphism $j_x : \mathrm{Spec}(\mathcal{O}_{X,x}) \to X$ is a homeomorphism onto its image. The image consists of those $y \in X$ which generalize $x$, i.e. satisfy $x \in \overline{\{y\}}$. Locally, this is just the classification of prime ideals in the localization.

It follows that that the image is closed under generalizations, and contains $x$. In general, I doubt that anything more can be said.

Every open subset is stable under generalizations. The converse is true for constructible subsets of noetherian sober topological spaces (see here). Chevalley's theorem implies that the image of a morphism of finite presentation is constructible. In particular, when $x$ is a closed point and $X$ is noetherian, the image of $j_x$ is open.