Does every non-empty topological space have an irreducible closed subset? So I have been reading through some threads on Mathstack about the irreducible components of a topological space, and it suddenly dawned on me:

How can we guarantee that any non-empty topological space actually contains an irreducible closed subset?

Ok so on the surface it would appear to make sense, as if it didnt contain an irreducible component you could "split" it into pieces indefinitely. But Maths can be strange, so maybe there is a case where you CAN do this? Or is it a fact: we MUST have a single point set {point} (which ofcourse would be irreducible).
As a side question: are there any zero dimensional topological spaces which are NON-discrete? The abundance of discrete topological spaces which this property is making me doubt that there is?
 A: Let $X$ be any space, let $x\in X$ be arbitrary, and let $F=\operatorname{cl}\{x\}$. Suppose that $F=H\cup K$, where $H$ and $K$ are closed. Without loss of generality $x\in H$, so $\{x\}\subseteq H$. But then $F=\operatorname{cl}\{x\}\subseteq H$, so $H=F$, and $F$ is irreducible. Thus, every space contains a non-empty irreducible closed set.
There are many non-discrete zero-dimensional spaces. For example, let $I$ be any infinite index set, and for each $\alpha\in I$ let $X_\alpha$ be a discrete space with at least two points; then $X=\prod_{\alpha\in I}X_\alpha$ is a zero-dimensional Hausdorff space that is not discrete. If each $X_\alpha$ is finite, $X$ is a product of compact spaces, so it is itself compact. When $I$ is countably infinite and each $X_\alpha$ is finite, $X$ is homeomorphic to the middle-thirds Cantor set.
Some other familiar examples are the following subspaces of the real line: $\{0\}\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$, $\Bbb Q$, and $\Bbb R\setminus\Bbb Q$. The last of these, like the Cantor set, is a product of discrete spaces: $\Bbb R\setminus\Bbb Q$ is homeomorphic to the product of countably infinitely many copies of $\Bbb N$. The first two, however, are not.
