The Iterative Simplicity of Basic Topological Operations

For convenience, we shall denote by c(A) and i(A) the closure, respectively, interior of a subset A ⊆ X.

As is well known, some of the iterates of c and i play an important role in topology. For instance, a subset A ⊆ X is called nowhere dense, iff i(c(A)) = φ. Further, a countable union of nowhere dense subsets is called of first Baire category, the essential fact in this regard being that no complete metric space is of first Baire category.

Motivated by the above, here various iterates of c and i, as well as of other related basic operations will be considered. In this regard we can note that all such operations are mappings of P(X) into itself, thus their compositions are associative. Consequently, we can consider the free semigroup S (T ) of mappings of P(X) into itself generated by these topological operations which will be listed below. This semigroup is in general obviously noncommutative.

The nontrivial aspect involved is that, in view of well known relations, such as

a number of elements in this noncommutative semigroup S (T ) correspond to the same mappings of P(X) into itself. Therefore, their identification, that is, the identification of the different elements in S (T ) is of interest.

Here we shall be interested in such identification which hold for all topological spaces (X, T ). Obviously, in the case of particular topological spaces, one can find more such identifications.

Let us list now the other mappings of P(X) which we shall consider.

One of them is the operation of taking the complementary, namely, ∁(A) = X \ A, with A ⊆ X. Here we have further reductions in the different elements of S (T ), since

One also defines the exterior e(A) of a subset A ⊆ X, given by

Another important topological operation is that of boundary of a subset A ⊆ X, which we shall denote by b(A). Here we have the obvious relation

or in view of (1.4)

Let us recall that a subset A ⊆ X is called a boundary set, iff

Let us now consider the related operations

called respectively the internal and external boundary of A. Clearly, we have

There are several further frequently used topological operations. One of them is the derived set d(A) of a subset A ⊆ X, defined by

And now we define (1.13) S (T )

as the free semigroup generated by the set of mappings {c, i, ∁, e, b, b i , b e , d}, and as customary with semigroups, we assume that it contains the neutral element id X which maps each A ⊆ X into itself.

Let us start with the simpler problem of studying the subsemigroup of S (T ) which is generated by the two operations c and i alone. This subsemigroup, in view of (1.1), is obviously given as follows

Let us consider the partial order relation α → β between mappings of P(X) into itself, defined by

On the other hand, we have

S c, i (T ) is a partially ordered semigroup with respect to →, and in general it is noncommutative.

Proof.

Let α, β, γ ∈ S c, i (T ), with α → β.

We show that γ α → γ β. Let A ⊆ X. Then (2.2) gives α(A) ⊆ β(A), hence (2.1), (2.4) result in γ(α(A)) ⊆ γ(β(A)).

Similarly, α γ → β γ. Indeed, for A ⊆ X, we have γ(A) ⊆ X, hence (2.2) gives α(γ(A)) ⊆ β(γ(A)).

We show that, in general, none of the relations holds

Thus the first and the second of the above relations do not hold. Let now A = [0, 1], then i(c(A)) = (0, 1) [0, 1] = c(i(A)), hence the third relation above cannot hold.

The relations hold

For the first relation in (2.6) we compose (2.3) on the left with c and obtain ci → c → c 2 , thus ci → c. Composing now (2.3) on the right with i, the result is i 2 → i → ci, or i → ci.

The second relation in (2.6) follows by composing (2.3) on the right with c, and thus obtaining ic → c → c 2 or ic → c. While composing (2.3) on the left with i, it follows that i 2 → i → ic, or i → ic.

For the third relation in (2.6) we compose the first relation in it with i on the left and obtain i 2 → ici → ic, and then recall the second relation in (2.6).

The fourth relation in (2.6) is obtained by composing the second relation in it with c on the left, with the result ci → cic → c 2 , and then use the first relation in (2.6).

The fifth relation in (2.6) results from the composition on the left with c of the third relation in (2.6), which gives ci → cici → cic → c 2 , after which we recall the first relation in (2.6).

The sixth relation in (2.6) comes from composing the fourth relation in (2.6) with i on the left, thus having i 2 → ici → icic → ic, and then recalling the second relation in (2.6).

For (2.7) we proceed as follows.

We have ci → cici from the fifth relation in (2.6). On the other hand, the fourth relation in (2.6) gives cic → c, which composed on the right with i, yields cici → ci.

As for icic → ic, it follows from the sixth relation in (2.6). Now the third relation in (2.6) gives i → ici, which composed with c on the right results in ic → icic.

Theorem 2.1.

The typically noncommutative semigroup generated by t

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