Example of uncountable infinite union of disjoint open sets in topology So I've been trying to look at the basics of Topology and in particular an analogy to do with computer science which is related to countability and the closure under arbitrary union property of Topologies. The details are irrelevant here, but it made me try and wonder what such a thing would actually look like. It always helps to have some first examples to work with when studying math like this!

Would anyone happen to be able to give some (as elementary as possible) examples of where an uncountable infinite union of open sets might show up?

Browsing Google didn't work. My first idea was to union intervals in $\mathbb{R}$, but then I basically stumbled across this realization for myself. Is there anything elementary that this works better with? I know that there must be, but I'm just not coming up with any at the moment. Thanks!
Oh also I thought to try it with open rectangles in the plane, but am rather suspicious about this idea for some reason.
 A: A nice space to illustrate this: the jungle river metric on $\Bbb R^2$.
Here the distance $d(x,y)$ is defined by imagining the $x$-axis as a river on whic we can travel and the erst of the plane is a large jungle, where we can only travel to the river vertically. Then to reach another point we "travel" along the river to the right vertical path and go up that. So $d(x,y) = |x_2- y_2|$ if $x_1 =y_1$ ($x,y$ on the same vertical path) and otherwise $(x_1 \neq y_1$) equals $|x_1 - y_1| + |x_2| + |y_2|$, where the last two terms are for getting to the river (for both points) and the first is the distance along the river.
One can check that this indeed is a valid metric, and all sets $N_x:=\{x\} \times (0, +\infty)$ (the northern path at $x$) and $S_x = \{x\} \times (-\infty,0)$ (the south path) are open for every $x \in \Bbb R$ (both are copies of $\Bbb R$, really) and clearly pairwise disjoint. So there you can see uncountably many disjoint open sets (non-empty and non-trivial) in a metric space. In a linear space like $\ell^\infty$ these are also easy to find, but this is a space which is more easily explained to beginning students, IMO.
Hope this helps your intuition..
A: The simplest example is with uncountably (in fact arbitrarily) many sets that are all the empty set. They are not distinct, but nevertheless they are disjoint!
With nonempty and disjoint, you need at least a space that does not have a countable dense subset (as is unfortunately the case with $\Bbb Q$ in $\Bbb R$ or $\Bbb Q^n$ in $\Bbb R^n$ or $\Bbb Q[X]$ in $C([0,1])$ -- all with their standard topologies).  So you can either make the spaces "bigger" or the open sets "smaller". For the latter, you can consider $\Bbb R$ with discrete topology and take the singleton sets as open sets. For the former, you can consider the long line, which is mostly like the real line, but so much longer that enough disjoint intervals fit in.
