Why do we define regular open/closed sets?

Question

I'd like to see some motivation or application of the definition of regular open/closed sets. There are answers already on this site that talk about the "geometric intuition" and the nice properties regular open sets have. However, the "interior of closure is itself" definition does not follow from results in $\mathbb{R}$ as others do(limit points, closure, limit of sequence, etc). I looked up on every single general topology book I know(Willard, Engelking, Kelley, Munkres...) but the only results about regular open/closed sets I can find is just about themselves(Like the "union" of two regular open sets, product of them, etc). If there is no applications of it, and no apparent motivations exist as examples(in $\mathbb{R}$, $\mathbb{R}^n$...), then why do we define such classes of sets in topological spaces?

Answer 1

Regular open sets can be a technical tool: e.g. we can show that in a regular space $X$ we have that $w(X)\le 2^{d(X)}$, where $w(X)$ is the minimal infinite cardinality of a base for $X$ and $d(X)$ is the minimal infinite cardinality of a dense subset of $X$. These are so-called cardinal invariants of topological spaces and their relations have been extensively studied, e.g. in metric spaces $w(X)=d(X)$, but in general $d(X) \le w(X)$ and the aforementioned inequality goes the other way, bounding $w(X)$ in terms of $d(X)$. For that proof we use that the regular open sets form a base for $X$ in a nice way.

But I think the main reason they are studied is the fact that $RO(X)$, the set of regular open sets of $X$, is a natural Boolean algebra, with operations $A \land B = A \cap B$, $A \lor B = \operatorname{int}(\overline{A \cup B)})$ and $\lnot A = X\setminus \overline{A}$ and $\emptyset, X$ as $0$ and $1$.

Another natural Boolean algebra asssociated with a space $X$ is $\operatorname{Clop}(X)$, the set of clopen (closed and open) subsets of $X$. which always contains $\emptyset,X$ (and in connected spaces only these two!) and where we use standard intersection, union and complement as Boolean operations.

One can show that if $B$ is any abstract Boolean algebra there is a compact Hausdorff space $X$ such that $B$ is isomorphic (as a Boolean algebra) to $\operatorname{Clop}(X)$ (and the clopen sets of $X$ form a base for its topology (such spaces are called zero-dimensional). Moreover, $RO(X)$ for that $X$ turns out to be a so-called completion of $B$ (it's clear that clopen sets are regular open so $\operatorname{Clop}(X) \subseteq RO(X)$) in which we can take arbitary $\lor,\land$ operations and all complete Boolean algebras are just regular open algebras of special compact spaces. So regular open sets are used in this connection to set theory/algebra and play an important role there.

Answer 2

The best motivation for doing anything in general topology is to classify topological spaces up to homeomorphism. In particular, words are defined so that when two spaces are not homeomorphic, we can say why in a sentence, e.g., "one is Hausdorff and the other isn't".

Regular open sets help in this regard. If $X$ is a regular space, then its regular open sets form a base for the topology on $X;$ but the converse statement is not true. There exist spaces which are semiregular, i.e., the regular open sets form a base for the topology, but which are not regular. And in turn, there exist semiregular spaces which are not locally regular.

Now imagine how cumbersome it would be to define "semiregular" without the notion of "regular open set", and then imagine how cumbersome it would be to read/write the previous paragraph if we hadn't defined "semiregular".