I was wondering about a problem related to regions in differentiable manifolds and I wanted to use an induction. Firstly I didn't notice that in an open set only could be countable connected components, so I was thinking to use an induction in a union with uncountable index.
My question is: Can you give me an example of a topological property that holds by countable union/intersection but not by uncountable union/intersection?
Thank you.
A set is meager if it is the countable union of nowhere dense sets. While meager-ness is preserved under countable unions (since countable $\times$ countable = countable), it is not preserved under arbitrary uncountable unions. For example, in $\mathbb{R}$ the whole space is not meager$^*$, but $$\mathbb{R}=\bigcup_{x\in\mathbb{R}}\{x\}$$ and every singleton is meager in $\mathbb{R}$.
$^*$This is not obvious! The Baire category theorem states that $\mathbb{R}$ with the usual topology, and many other spaces, is not meager. The Baire category theorem has many applications; I don't think I can write a good summary here, but if you google around (and search on this site) you'll find many uses of it.
Relatedly, there is a rich hierarchy of "types of set" beyond just open and closed. For example, we inductively define the classes $\Sigma_\alpha^0$ and $\Pi^0_\alpha$ for $\alpha$ a countable ordinal) as:
$\Sigma^0_1$= open, $\Pi^0_1$= closed.
$\Sigma^0_\alpha$ is the set of countable unions of sets, each of which is $\Pi^0_\beta$ for some $\beta<\alpha$; and $\Pi^0_\alpha$ is the set of countable intersections of sets, each of which is $\Sigma^0_\alpha$ for some $\beta<\alpha$.
(Note that $X$ is $\Sigma^0_\alpha$ iff the complement of $X$ is $\Pi^0_\alpha$. A set is $\Delta^0_\alpha$ if it is both $\Sigma^0_\alpha$ and $\Pi^0_\alpha$.)
This hierarchy - the Borel hierarchy - makes sense for arbitrary spaces. For $\mathbb{R}$ with the usual topology, and related spaces, it is particularly nice: for example, in $\mathbb{R}$ with the usual topology we have that every $\Sigma^0_\alpha$ (resp. $\Pi^0_\alpha$) set is $\Sigma^0_\gamma$ (resp. $\Pi^0_\gamma$) whenever $\alpha<\gamma$, and this hierarchy is strict ($\Sigma^0_\gamma$ properly contains $\Sigma^0_\alpha$ for $\gamma>\alpha$). We can of course go beyond the Borel hierarchy (for example, via the projective hierarchy) but already the Borel hierarchy contains basically every set of reals you can easily explicitly describe.
Each level of the Borel hierarchy has some amount of closure: e.g. $\Sigma^0_\alpha$ is closed under countable unions but not intersections and $\Pi^0_\alpha$ is closed under countable intersections but not unions. It's a good exercise to show that "countable" is necessary here in general (think about $\alpha=1$). So these provide additional examples.
Incidentally, you may find topics in descriptive set theory of particular interest; Kechris' book is an excellent introduction.