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!?

How about this:
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. 
