Irreducible subsets of a topological space

I found this definition on Hartshorne, Algebraic geometry, page 3...

Definition A nonempty subset $Y$ of a topological space $X$ is irreducible if it cannot be expressed as the union $Y=Y_1\cup Y_2$ of two proper subsets, each one of which is closed in $Y$. The empty set is not considered to be irreducible.

I suppose this definition is made for algebraic sets and Zariski topology, but I was wondering if it could be applied to different contexts...For instance, take $X:=\mathbb{R}^2$ with the standard topology. Let $Y$ be a line, for example the $x$-axis. So $Y$ is reducible (i.e. not irreducible) because for example $Y=(-\infty,1]\cup [0,+\infty)$. First question: does it make sense what i just said?

It seems to me that in the standard topology on $\mathbb{R}^n$ "everything" is reducible in a trivial way, and this seems to me very strange. So could you provide some examples of irreducible subsets of a not-too-exotic topological space?

A minor question: why is it made explicit the fact that the empty set is not irreducible? I mean, the empty set has no proper subsets, or not!?

• For example a topological space with only a point. – Dubious Oct 14 '13 at 18:48

In the language of general topology (i.e. not algebraic geometry) such a space is more commonly known as hyperconnected, though either term is acceptable. Intuitively, $\mathbb R$ with its standard topology has far too many open sets to be hyperconnected/irreducible. For a set to be hyperconnected its open sets need to be "large", in the sense that every pair of nonempty open sets needs to have nonempty intersection (this is an equivalent definition to the one Hartshorne uses). Of course, that automatically rules out any Hausdorff spaces (except fairly trivial cases) so examples in terms of "ordinary" things like metric spaces are impossible.
The cofinite topology on any infinite set is an example of a hyperconnected space, and this is probably a better thing to think about intuitively to get a grasp on the Zariski topology. Indeed, the Zariski topology on $\mathbb A^1$ agrees with the cofinite topology, but in larger dimensions the Zariski topology is somewhat larger than that but still fairly small compared to most of the spaces you are used to.
Such sets are also called hyperconnected, because they are connected in a very strong sense: the definition of irreducible is equivalent to saying that if $U$ and $V$ are non-empty open subsets of the set, then their intersection is non-empty.
A nice example of a hyperconnected space is any infinite set $X$ with the cofinite topology: $U\subseteq X$ is open if and only if $X\setminus U$ is finite, or $U=\varnothing$. In terms of separation axioms this is about as nice as a hyperconnected space can get: it’s $T_1$, and clearly no hyperconnected space with more than one point can be Hausdorff.
Generate a topology on $\mathbb{R} \times \mathbb{R}$ by defining closed sets to be the horizontal and vertical lines. (This gives a subbasis whose open sets are complements of lines; therefore all finite intersections of said complements yield a topology.) The closed sets are lines, finite collections of parallel lines, and finite point sets. While the latter two types of closed sets are not irreducible, the first type, the lines, are, since the only closed set properly contained in a line will be finite point sets and no finite union of finite point sets yields a line.