Why are the irreducible components of a subspace of Noetherian space just the intersection with the irreducible components?

Let $X$ be any topological space. The irreducible components of $X$ meeting $U$ correspond to the irreducible components of any open subset $U$ of $X$, via intersecting with $U$.

In this paragraph, we will give a bijection between irreducible closed subsets of $X$ meeting $U$ and the irreducible closed subsets of $U$.

• If $Z$ is an irreducible closed subset of $X$ meeting $U$, then $Z\cap U$ is irreducible. This is obvious because a nonempty space is irreducible if and only if any two nonempty open subspaces have nonempty intersection, by definition.
• If $T = Z\cap U$ is irreducible, where $Z\subset X$ closed, then the closure $\overline{T}$ of $T$ in $X$ is irreducible. This is obvious because the closure of an irreducible set is irreducible as well, by Exercise 3.6.B(b).
• If $Z$ is an irreducible closed subset of $X$ meeting $U$, then the closure of $Z\cap U$ in $X$ is just $Z$. This is true because in an irreducible topological space, any nonempty open subspace is dense.
• If $T = Z\cap U$ irreducible, where $Z\subset X$ closed, then $\overline{T}\cap U = T$, where $\overline{T}$ is the closure of $T$ in $X$. On one hand, $\overline{T} = \overline{Z\cap U}\subset Z$ and thus $\overline{T}\cap U\subset Z\cap U = T$. On the other hand, $T\subset\overline{T}$ and $T\subset U$ and thus $T\subset\overline{T}\cap U$.

In this paragraph, we will show that the bijection above maps components to components in both directions.

• If $Z$ is an irreducible component of $X$ meeting $U$, then $Z\cap U$ is an irreducible component of $U$. Let $S$ be an irreducible closed subset of $U$ containing $Z\cap U$. Then, $\overline{S}$ is an irreducible closed subset of $X$ containing $\overline{Z\cap U} = Z$. Since $Z$ is an irreducible component, $\overline{S} = Z$. Thus, $S = \overline{S}\cap U = Z\cap U$.
• If $T$ is an irreducible component of $U$, then $\overline{T}$ is an irreducible component of $X$. Let $Y$ be an irreducible closed subset of $X$ containing $\overline{T}$. Then, $Y\cap U\supset\overline{T}\cap U = T$. Since $T$ is an irreducible component of $U$, $Y\cap U = T$. Thus, $Y = \overline{Y\cap U} = \overline{T}$.

Replacing $X$ by $X_i$ and $U$ by $U \cap X_i$ you only have to prove that for any irreducible space $X$ and any nonempty open subset $U \subseteq X$, $U$ is also irreducible. If $U = (U \cap Z_1) \cup (U \cap Z_2)$ for some closed subsets $Z_1$, $Z_2$ of $X$, then the closure $\overline{U}$ of $U$ in $X$ is $\overline{U \cap Z_1} \cup \overline{U \cap Z_2}$. As $U$ is a nonempty open subset of an irreducible space, in particular dense in $X$, this implies $\overline{U \cap Z_1} \cup \overline{U \cap Z_2} = X$ and irreducibility gives us (say) $X = \overline{U \cap Z_1}$. Intersecting with $U$ now gives $U = U \cap Z_1$.