Proposition

Let $X$ be a topological space.

If $Y$ is an irreducible subspace of $X$, then the closure $\overline{Y}$ of $Y$ in $X$ is irreducible.
Every irreducible subspace of $X$ is contained in a maximal irreducible subspace.
The maximal irreducible subspaces of $X$ are closed and cover $X$. What are the irreducible components of a Hausdorff space?
TODO

Solution

1

Since $Y$ is nonempty, $\overline{Y}$ is nonempty.
Let $U, V$ be nonempty open subsets of $\overline{Y}$. Then $U \cap Y$ and $V \cap Y$ are open subsets of $Y$. Suppose $U \cap Y = \emptyset$. Then $\overline{Y} \setminus U$ is a closed set containing $Y$. This is impossible because $\overline{Y}$ is the smallest closed set containing $Y$. Thus $U \cap Y \ne \emptyset$. Similarly, $V \cap Y \ne \emptyset$. Thus $(U \cap Y) \cap (V \cap Y)$ is nonempty by the irreducibility of $Y$. Therefore, $U \cap V \ne \emptyset$.

Therefore, $\overline{Y}$ is irreducible.

2

Let $A$ be an irreducible subspace of $X$. Let $\Sigma$ be the set of all irreducible subspaces of $X$ containing $A$. Let $Y_1 \subset Y_2 \subset \cdots$ be a chain in $\Sigma$. Let $Y = \bigcup Y_i$. We claim that $Y \in \Sigma$. Let $U, V$ be nonempty open subsets of $Y$. Let $x \in U, y \in V$. Then $x \in Y_i$ and $y \in Y_j$ for some $i, j$. Then $x, y \in Y_k$ where $k = \max(i, j)$. Then $U \cap Y_k \ne \emptyset$ and $V \cap Y_k \ne \emptyset$. Since $U \cap Y_k$ and $V \cap Y_k$ are both open in $Y_k$ and $Y_k$ is irreducible, $(U \cap Y_k) \cap (V \cap Y_k) \ne \emptyset$. Thus $U \cap V \ne \emptyset$.

By Zorn’s Lemma, $\Sigma$ contains a maximal element. In other words, there exists a maximal irreducible subspace of $A$.

3

Let $Y$ be a maximal irreducible subspace of $X$. By 1, $\overline{Y}$ is irreducible. Since $Y$ is maximal, $Y = \overline{Y}$, so $Y$ is closed.

Every singleton $\{ x \} \subset X$ is clearly irreducible. By 2, every singleton is contained in a maximal irreducible subspace. Thus the maximal irreducible subspaces cover $X$.

The singletons are the irreducible components of a Hausdorff space.

4

TODO