## Regular open sets

**Exercise**
Let *A* be a subset of a topological space *X*.
How many different subsets can I make starting from *A* and
repeatedly using the closure and interior operations?
Answer: 7.

Let us write o for interior and - for closure, so that
the closure of the interior of *A* is written *Ao-*.
Now the seven sets are *A*, *A-*, *Ao*, *A-o*,
*Ao-*, *A-o-*, *Ao-o*.
It stops here because *A-o-o = A-o* and *Ao-o- = Ao-*.

**Proof**
Since *A-o* is open, it is contained in *A-o-o*.
Since *A-* is closed it contains *A-o-*, and hence
*A-o* contains *A-o-o*.

## The Boolean algebra of regular open sets

A subset *A* of a topological space *X*
is called *regular open* when *A = A-o*.
The regular open sets of a topological space form
a Boolean algebra with operations

**Exercise**
Check this.

## Galois Correspondence

Suppose *P* and *Q* are partially ordered sets
and *f:P->Q* and *g:Q->P* order reversing maps.
Assume *fgf >= f* and *gfg >= g*. Then *gfgf = gf*.
It follows that *f* and *g* set up a 1-1 correspondence
between *gf[P]* and *fg[Q]*.
**Proof**
The first inequality yields *gfgfp <= gfp* for all *p*,
and the second one *gfgfp >= gfp* for all *p*.

Of course the same conclusion holds in case *fgf <= f*
and *gfg <= g*.

**Example**
In a topological space *X* take for both *P* and *Q*
the power set of *X* ordered by inclusion, and take
*f = g = -c* where *c* is taking complements
and *-* is the closure operator.
Indeed, note that *c-c = o* the interior operator, and
*A-o- <= A-* so that the hypothesis is satisfied.
The conclusion says *-o-o = -o* and *o-o- = o-*.

## Simple Galois Correspondence

Suppose *P* and *Q* are partially ordered sets
and *f:P->Q* and *g:Q->P* order reversing maps.
Assume *gf >= 1* and *fg >= 1*, where *1* is
the identity mapping. Then *fgf = f* and *gfg = g*.
If follows that *f* and *g* set up a 1-1 correspondence
between *g[Q]* and *f[P]*.
**Example**
Let *P* be the collection of subfields of a field *F*
and let *Q* be the collection of subgroups of the group
*G = Aut F*. Define *f(K)* for a subfield *K*
of *F* as the group of automorphisms of *F* that fix
*K* pointwise. Define *g(H)* for a subgroup *H*
of *G* as the subfield fixed by *H*.
This is the original Galois correspondence.

Évariste Galois