Kuratowski monoids of $n$-topological spaces

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $\tau_1\subset\dots\subset\tau_n$, by repeated application of the operations of complement and closure in the topologies $\tau_1,\dots,\tau_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2\sum_{i,j=0}^n\binom{i+j}{i}\binom{i+j}{j}$ distinct sets.


Introduction
This paper was motivated by the famous Kuratowski 14-set closure-complement Theorem [1], which says that the repeated application of the operations of closure and complement to a subset A of a topological space X yields at most 14 pairwise distinct sets 1 . More precisely, this theorem says that for any topological space (X, τ ) the operators of complement c : P(X) → P(X), c : A → X \ A and closureτ : P(X) → P(X), τ : A →Ā, generate a submonoid c,τ of cardinality ≤ 14 in the monoid P(X) P(X) of all self-maps of the power-set P(X) of X.
In [4] Shallitt and Willard constructed two commuting closure operators p, q : P(X) → P(X) on the powerset P(X) of a countable set X such that the submonoid p, q, c ⊂ P(X) P(X) generated by these closure operators and the operator of complement is infinite. In Example 3.1 below we shall define two metrizable topologies τ 1 and τ 2 on a countable set X such that the closure operatorsτ 1 andτ 2 in the topologies τ 1 and τ 2 generate an infinite submonoid τ 1 ,τ 2 in the monoid P(X) P(X) of all self-maps of P(X). Moreover, for some set A ⊂ X the set {f (A) : f ∈ τ 1 ,τ 2 } is infinite. This shows that Kuratowski's 14-set theorem does not generalize to spaces endowed with two or more topologies.
The situation changes dramatically if two topologies τ 1 and τ 2 on a set X are comparable, i.e., one of these topologies is contained in the other. In this case we shall prove that the closure operatorsτ 1 ,τ 2 : P(X) → P(X) induced by these topologies together with the operator c of complement generate a submonoid τ 1 ,τ 2 , c ⊂ P(X) of cardinality ≤ 126. In fact, we shall consider this problem in a more general context of multitopological spaces and polytopological spaces.
By a multitopological space we understand a set X endowed with a family T of topologies on X. A multitopological space (X, T ) is called polytopological if the family of its topologies T is linearly ordered by the inclusion relation. A typical example of a polytopological space is the real line endowed with the Euclidean and Sorgenfrey topologies. Another natural example of a polytopological space is any Banach space, carrying the norm and weak topologies. A dual Banach space is an example of a polytopological space carrying three topologies: the norm topology, the weak topology and the * -weak topology. A topological space (X, τ ) can be thought as a polytopological space (X, {τ }) endowed with the family {τ } consisting of a single topology τ .
For a topology τ on a set X byτ : P(X) → P(X) andτ : P(X) → P(X) we shall denote the operators of taking the interior and closure with respect to the topology τ . These operators assign to each subset A ⊂ X its interiorτ (A) and closureτ (A), respectively. Since τ = {τ (A) : A ⊂ X} = {X \τ(A) : A ⊂ X} the topology τ can be recovered from the operatorsτ andτ .
For a multitopological space X = (X, T ) the submonoid in P(X) P(X) generated by the interior and closure operatorsτ ,τ for τ ∈ T , will be called the Kuratowski monoid of the multitopological space X. A somewhat larger submonoid K 2 (X) = c,τ : τ ∈ T in P(X) P(X) generated by the operator of complement c and the closure operatorsτ , τ ∈ T , will be called the full Kuratowski monoid of the multitopological space X.
The notion of a multitopological space has one disadvantage: multitopological spaces do not form a category (it is not clear what to understand under a morphism of multitopological spaces). This problem with multitopological spaces can be easily fixed by introducing their parametric version called L-topological spaces where (L, ≤) is a partially ordered set.
Given a subset X we denote by Top(X) the family of all possible topologies on X, partially ordered by the inclusion relation. The family Top(X) is a lattice whose smallest element is the anti-discrete topology τ a and the largest element is the discrete topology τ d on X. Observe that for the discrete topology the operatorsτ d andτ d coincide with the identity operator 1 X on P(X).
Let (L, ≤) be a partially ordered set. By definition, an L-topology on a set X is any monotone map τ : L → Top(X). The monotonicity of τ means that for any elements i ≤ j in L we get τ (i) ⊂ τ (j). In the sequel for an element i ∈ L it will be convenient to denote the topology τ (i) by τ i . By an L-topological space we shall understand a pair (X, τ ) consisting of a set X and an L-topology τ : L → Top(X) on X.
By a morphism between two L-topological spaces (X, τ ) and (Y, σ) we understand a map f : X → Y which is continuous as a map between topological spaces (X, τ i ) and (Y, σ i ) for every i ∈ L. L-Topological spaces and their morphisms form a category called the category of L-topological spaces. Each L-topological space X = (X, τ ) can be thought as a multitopological space endowed with the family of topologies {τ i } i∈L . If the set L is linearly ordered, then the multitopological space (X, {τ i } i∈L ) is polytopological.
So we can speak about the Kuratowski monoid K(X) and the full Kuratowski monoid K 2 (X) of an Ltopological space X.
We shall prove that the upper bound for the cardinality of the Kuratowski monoid K(X) of an n-topological space X is given by the number where n i = n! i!(n−i)! is the binomial coefficient. The main result of this paper is the following theorem. Theorem 1.1. For any n-topological space X = (X, T ) its Kuratowski monoid K(X) has cardinality |K(X)| ≤ K(n) and its full Kuratowski monoid K 2 (X) has cardinality |K 2 (X)| ≤ 2 · K(n).

The Kuratowski monoid of a saturated polytopological space
In this section we introduce a class of n-topological spaces X = (X, T ) whose Kuratowski monoids K(X) have cardinality strictly smaller than K(n).
A multitopological space (X, τ ) is called saturated if for any topologies τ 0 , τ 1 ∈ T each non-empty open subset U ∈ τ 0 has non-empty interior in the topology τ 1 . A typical example of a saturated multitopological space is the real line R endowed with the 2-element family T = {τ 0 , τ 1 } consisting of the Euclidean and Sorgenfrey topologies τ 0 ⊂ τ 1 .
For a linearly ordered set L an L-topological space (X, τ ) is defined to be saturated if the multitopological space (X, {τ i } i∈L ) is saturated.
Proof. The definition of a saturated polytopological space X = (X, T ) implies thatτ 0τ1 =τ 0τ0 for any topologies τ 0 , τ 1 ∈ T . Applying to this equality the operator c of taking complement, we get
Proof. The upper bound |K(X)| ≤ 13 follows from Theorem 2.1. To prove the lower bound |K(X)| ≥ 13, consider the subset and observe that the following 13 subsets of R are pairwise distinct, witnessing that |K(X)| ≥ |{f Theorem 2.1 gives a partial answer to the following general problem. Problem 2.3. Which properties of a polytopological space X are reflected in the algebraic structure of its Kuratowski monoid K(X)?
3. An example of a multitopological space with infinite Kuratowski monoid In this section we shall construct the following example announced in the introduction.
Example 3.1. There is a countable space X endowed with two (incomparable) metrizable topologies τ 0 , τ 1 such that the Kuratowski monoid K(X) of the multitopological space X = (X, Proof. Take any countable metrizable topological space X containing a decreasing sequence of non-empty subsets (X n ) n∈ω such that X 0 = X, n∈ω X n = ∅ and X n+1 is nowhere dense in X n for all n ∈ ω.

Kuratowski monoids
To prove Theorem 1.1 we shall use the natural structure of partial order on the monoid P(X) P(X) . For two maps f, g ∈ P(X) P(X) we write f ≤ g if f (A) ⊂ g(A) for every subset A ⊂ X. This partial order turns P(X) into a partially ordered monoid.
By a partially ordered monoid we understand a monoid M endowed with a partial order ≤ which is compatible with the semigroup operation of M in the sense that for any points x, y, z ∈ M the inequality x ≤ y implies xz ≤ yz and zx ≤ zy. Recall that a monoid is a semigroup S possessing a two-sided unit 1 ∈ S.
Observe that for two comparable topologies τ 1 ⊂ τ 2 on a set X we get where 1 X : P(X) → P(X) is the identity transformation of P(X). Now we see that for a polytopological space X = (X, T ) its Kuratowski monoid K(X) = τ ,τ : τ ∈ T is generated by the linearly ordered set This leads to the following A Kuratowski monoid K is called a Kuratowski monoid of type (n, p) if K has a linear generating set L such that |L − | = n and |L + | = p.
For two numbers n, p ∈ ω consider the number and observe that K(n) = K(n, n) for every n ∈ ω.
It is easy to see that for each polytopological space X = (X, T ) endowed with n = |T | topologies, its Kuratowski monoid K(X) is a Kuratowski monoid of type (n, n) or (n − 1, n − 1). The latter case happens if the polytopology T of X contains the discrete topology τ d on X. In this caseτ d = 1 X =τ d .
Now we see that Theorem 1.1 is a partial case of the following more general theorem, which will be proved in Section 7 (more precisely, in Theorem 7.1).
The values of the double sequence K(n, p) for n, p ≤ 9 were calculated by computer: By a pointed linearly ordered set we understand a linearly ordered set L with a distinguished element 1 ∈ L called the unit of L. This element divides the set L \ {1} into negative and positive parts L − = {x ∈ L : x < 1} and L + = {x ∈ L : x > 1}, respectively. By F S L = ∞ n=1 L n we denote the free semigroup over L. It consists of non-empty words in the alphabet L. The semigroup operation on F S L is defined as the concatenation of words. The set L is identified with the set L 1 of words of length 1 in the alphabet L.
A word w = x 1 . . . x n ∈ F S L of length n is called alternating if for each natural number i with 1 ≤ i < n the doubleton {x i , x i+1 } intersects both sets L − and L + . According to this definition, words of length 1 also are alternating. On the other hand, an alternating word of length ≥ 2 does not contain a letter equal to 1.
An alternating word are strictly increasing in L − and the sequences ( are strictly decreasing in L + and the sequences ( Let us calculate the cardinality of the set K L depending on the cardinalities n = |L − | and p = |L + | of the negative and positive parts of L.
For non-negative integers n, r by n r we denote the cardinality of the set of r-element subsets of an n-element set. It is clear that otherwise.
The numbers n r will be called binomial coefficients. The following properties of binomial coefficients are well-known (see, e.g. [3, §5.1]).
Lemma 5.1. For any non-negative integer numbers m, n, k we get (1) n k = n n−k , In the following theorem we calculate the cardinality of the set K L of Kuratowski words.
The number of such sequences is equal to a r where the cardinality a = |A| can vary from 0 (if x 2m is the largest element of the set L − ) till n − 1 (if x 2m is the smallest element of L − ). By analogy, (x 2m−2i ) 1≤i≤ 2m−k 2 is a strictly increasing sequence of length l = ⌊ 2m−k 2 ⌋ in the linearly ordered set A and the number of such sequences is equal to a l .
is a strictly decreasing sequence of length is a strictly decreasing sequence of length in the linearly ordered set B. The number of such sequences is equal to b r . Summing up and applying Lemma 5.1(2), we conclude that the family V ∓ of all V ∓ -words has cardinality  3. To calculate the number of W + -words, fix any W + -word w ∈ W + and write it as an alternating word w =

By analogy we can prove that
The number of such sequences is equal to a l where a = |A| < n. By analogy, (x 2m+2i ) 0≤i≤ q−2m 2 is a strictly increasing sequence of length r = ⌊ q−2m 2 ⌋ in the linearly ordered set A and the number of such sequences is equal to a r .
is a strictly decreasing sequence of length is a strictly decreasing sequence of length  6. An asymptotics of the sequence K(n)

By the preceding items
In this section we study the asymptotical growth of the sequence K(n) = K(n, n) and prove Theorem 1.3 announced in the Introduction as a corollary of the following results.

Now we have that
according to Lemma 6.3.

Representing elements of Kuratowski monoids by Kuratowski words
Let K be a Kuratowski monoid with linear generating set L and let L − = {x ∈ L : x < 1} and L + = {x ∈ L : x > 1} be the negative and positive parts of L, respectively. Let F S L = ∞ n=1 L n be the free semigroup over L and π : F S L → K be the homomorphism assigning to each word x 1 . . . x n ∈ F S L the product x 1 · · · x n of its letters in K. The homomorphism π : F S L → K induces a congruence ∼ on F S L which identifies two words u, v ∈ F S L iff π(u) = π(v).
A word w ∈ F S L is called irreducible if w has the smallest possible length in its equivalence class [w] ∼ = {u ∈ F S L : u ∼ w}. Since the set of natural numbers is well-ordered, for each element x ∈ K there is an irreducible word w ∈ F S L such that x = π(w). Consequently, the cardinality of K does not exceed the cardinality of the set of irreducible words in F S L .
Theorem 7.1. Each irreducible word in F S L is a Kuratowski word. Consequently, π(K L ) = K and |K| ≤ |K L |. If the set L is finite, then |K| ≤ |K L | = K(n, p) where n = |L − | and p = |L + |.
Proof. We divide the proof of Theorem 7.1 into a series of lemmas.
Proof. Since L is linearly ordered, either x ≤ y or y ≤ x.
If x ≤ y, then multiplying this inequality by x, we get x = xx ≤ xy. On the other hand, multiplying the inequality y ≤ 1 by x, we get the reverse inequality xy ≤ x1 = x. Taking into account that x ≤ xy ≤ x, we conclude that x = xy.
If y ≤ x, then multiplying this inequality by y, we get y = yy ≤ xy. On the other hand, multiplying the inequality x ≤ 1 by y, we get xy ≤ 1y = y. Taking into account that y ≤ xy ≤ y, we conclude that xy = y = min{x, y}.
By analogy we can prove: Lemma 7.3. For any elements x, y ∈ L + ∪ {1} we get xy = max{x, y}.
Proof. If x ≤ y, then multiplying this inequality by y, we obtain xy ≤ yy = y. On the other hand, multiplying the inequality 1 ≤ x by y, we get y = 1y ≤ xy. So, xy = y = max{x, y}.
If y ≤ x, then after multiplication by x, we obtain xy ≤ xx = x. On the other hand, multiplying the inequality 1 ≤ y by x, we get x ≤ xy and hence xy = x = max{x, y}.
Recall that a word x 1 . . . x n ∈ F S L is alternating if for each natural number i with 1 ≤ i < n the doubleton {x i , x i+1 } intersects both sets L − and L + . According to this definition, one-letter words also are alternating. Lemmas 7.2 and 7.3 imply: Lemma 7.4. Each irreducible word w ∈ F S L is alternating.
The following lemma will help us to reduce certain alternating words of length 4.
Lemma 7.5. If x 1 x 2 x 3 x 4 ∈ F S L is an alternating word in the alphabet L such that Proof. Two cases are possible.
1) x 1 , x 3 ∈ L − and x 2 , x 4 ∈ L + . In this case the equalities x 1 x 3 = x 1 and x 2 x 4 = x 4 imply that x 1 ≤ x 3 and x 2 ≤ x 4 (see Lemmas 7.2 and 7.3). To see that On the other hand, 2) x 1 , x 3 ∈ L + and x 2 , x 4 ∈ L − . In this case the equalities x 1 x 3 = x 1 and x 2 x 4 = x 4 imply that x 1 ≥ x 3 and x 2 ≥ x 4 (see Lemmas 7.2 and 7.3). To see that On the other hand, Now we are able to prove that each irreducible word w ∈ F S L is a Kuratowski word. If w consists of a single letter, then it is trivially Kuratowski and we are done. So, we assume that w has length ≥ 2. By Lemma 7.4 the word w is alternating and hence can be written as the product w = x k · · · x n for some k ∈ {0, 1} and n > k such that x 2i ∈ L − for all integer numbers i with k ≤ 2i ≤ n, and x 2i−1 ∈ L + for all integer numbers i with k ≤ 2i − 1 ≤ n.
Let m be the smallest number such that k ≤ 2m ≤ n and x 2m = min{x 2i : k ≤ 2i ≤ n} in L − . First we shall analyze the structure of the subword x k · · · x 2m of the word w = x k . . . x n .
Next, we consider the subword x 2m+1 · · · x n of the word w = x k · · · x n .
Therefore each irreducible word in F S L is a Kuratowski word, which implies that |K| ≤ |K L |. If the set L is finite, then the set K L of Kuratowski words over L has cardinality |K L | = K(|L − |, |L + |), see Theorem 5.2.

Separation of Kuratowski words by homomorphisms
In the preceding section we proved that any element of a Kuratowski monoid K with a linear generating set L can be represented by a Kuratowski word w ∈ K L . In this section we shall prove that Kuratowski words can be separated by homomorphisms into the Kuratowski monoids of suitable 2-topological spaces.
Given an n-topological space X = X, (τ i ) i∈n , observe that the linear generating set This observation motivates the following definition. A * -linearly ordered set is a linearly ordered set L endowed with an involutive bijection * : L → L, * : ℓ → ℓ * , that has a unique fixed point 1 ∈ L and is decreasing in the sense that for any elements x < y in L we get x * > y * . Each * -linearly ordered set L is pointed -the unit of L is the unique fixed point of the involution * : L → L. Observe that the structure of a * -linearly ordered set L is determined by the structure of its negative part L − . A map f : L → Λ between two * -linearly ordered sets L, Λ will be called a * -morphism if f is monotone (in the sense that for any elements x ≤ y of L we get f (x) ≤ f (y)) and preserves the involution (in the sense that f (x * ) = f (x) * for every x ∈ L. Since f (1) = f (1 * ) = f (1) * , the image f (1) of the unit of L coincides with the unit of Λ). Observe that each * -morphism f : L → Λ is uniquely determined by its restriction f |L − .
For a * -linearly ordered set L, the involution * : L → L of L has a unique extension to an involutive semigroup isomorphism * : F S L → F S L of the free semigroup over L. The image of a word w ∈ F S L under this involutive isomorphism will be denoted by w * .
Let X = (X, T ) be a polytopological space and be the linear generating set of the Kuratowski monoid K(X) of X. Observe that each topology τ is determined by its interior operatorτ (since τ = {τ (A) : A ⊂ X}). This implies that the interior operatorsτ , τ ∈ T , are pairwise distinct. The same is true for the closure operatorsτ , τ ∈ T . This allows us to define a bijective involution * : L(X) → L(X) lettingτ * =τ andτ * =τ for every τ ∈ T . This involution turns L(X) into a * -linearly ordered set. Let L be a * -linearly ordered set. Choose any point c / ∈ L and consider the free semigroup F S L∪{c} over the set L ∪ {c}. This semigroup consists of words in the alphabet L ∪ {c}. Let X = (X, T ) be a polytopological space and L(X) be the linear generating set of the Kuratowski monoid K(X) of X. Let c X : P(X) → P(X), c X : A → X \ A, denote the operator of taking complement.
Observe thatf (K L ) ⊂ K(X). In the semigroup F S L∪{c} consider the subset K L = K L ∪ {cw : w ∈ K L } ⊂ F S L∪{c} whose elements will be called full Kuratowski words.
Theorem 8.1. For any * -linearly ordered set L and any two distinct words u, v ∈ K L there is a 2-topological space X, and a * -morphism f : L → L(X) whose Kuratowski extensionf : F S L∪{c} → K 2 (X) separates the words u, v in the sense thatf (u) =f (v).
Proof. In most of cases the underlying set of the 2-topological space X will be a set X = {x, y} containing two pairwise distinct points x, y and the topologies of X are equal to one of four possible topologies on X: • τ d = ∅, {x}, {y}, X , the discrete topology on X; • τ a = ∅, X , the anti-discrete topology on X; Fix any two distinct words u, v ∈ K L and consider four cases. 1) u ∈ K L and v / ∈ K L . In this case consider the 2-topological space X = X, (τ a , τ d ) . Then for the * -morphism f : 2) v ∈ K L and u / ∈ K L . In this case take the 2-topological space X from the preceding case and observe that for the * -morphism f : 3) u, v ∈ K L . Denote by u 0 , v 0 ∈ L the last letters of the words u, v, respectively. Consider two cases. 3a) u 0 = v 0 . We lose no generality assuming that u 0 < v 0 (in the linearly ordered set L). 3ac) u 0 ∈ L − and v 0 ∈ L + . In this case, take the 2-topological space X, the subset A = {x} and the * -morphism f : L → L(X) from case (3aa). Thenû(A) = ∅ = X =v(A), which implies thatf (u) =f (v).
Next, consider the subcase: 3bab) v k+1 = v k−1 . In this case v ∈ W + and hence the sequences are strictly decreasing in L + . Consider the 2-topological space X = {x, y}, (τ a , τ y ) and define a * -morphism f : L → L(X) assigning to each element ℓ ∈ L − the operator Also consider the subset A = {x} in the 2-topological space X.
Finally, consider the subcase: is strictly decreasing in L + , we conclude that v k−1 > v k−1+2i for any non-zero integer number i with 0 ≤ k − 1 + 2i ≤ q and u k < min{u k−2 , v k } = min{v k−2 , v k } ≤ v k+2i for any integer number i with k + 2i ∈ {0, . . . , q}. Take the 2-topological space X, the subset A = {x}, and the * -morphism f : L → L(X) from the case (3bab). By analogy with the preceding case it can be shown thatv(A) = A = ∅ =û(A) and hencef (u) =f (v). This completes the proof of case (3ba).
So, now we consider the case: 3bb) u k ≥ u k−2 . This case is more difficult and requires to consider a set X = {x, y, z} of cardinality |X| = 3 endowed with the topologies: • τ The inequality u k−2 ≤ u k ∈ L − and the inclusion u ∈ K L imply that the sequence u k−1+2i 0≤i≤ p−k+1 2 is strictly decreasing in L + . By analogy, the (strict) inequality v k−2 = u k−2 ≤ u k < v k implies that sequence v k−2+2i 0≤i≤ q−k+2 2 is strictly increasing in L − and v k−1+2i 0≤i≤ q−k+1 2 in strictly decreasing in L + . Let u * k−1 be the letter in L − , symmetric to the letter u k−1 with respect to 1. Depending on the relation between u * k−1 and u k we consider two subcases: 3bba) u * k−1 ≤ u k . In this case consider the 2-topological space X = X, (τ x,z , τ x,z,yz ) where X = {x, y, z} and define the * -morphism f : L → L(X) assigning to every element ℓ ∈ L − the operator Consider the subset A = {x} of X. Observe that for every ℓ ∈ L we get {x} ⊂l({x}) ⊂l({x, y}) ⊂ {x, y}. Then ⊂ {τ x,z,yz , 1 X } and thenû(A) =û p · · ·û k+1 (A) ⊂ A. On the other hand, v k > u k implies thatv k · · ·v 1 (A) =v k ({x, y}) = 1 X ({x, y}) = {x, y}. Taking into account that u k < v k and the sequence (v k+2i ) 0≤i≤ q−k 2 is strictly increasing in L − , we conclude that 3bbb) u k < u * k−1 . In this case consider the 2-topological space X = X, (τ x,z , τ x,z,xy ) where X = {x, y, z}. Define a * -morphism f : L → L(X) assigning to every element ℓ ∈ L − the operator This completes the proof of case (3) under the assumption u k−1 = v k−1 ∈ L + . If u k−1 = v k−1 ∈ L − , then we can consider the dual words u * = u * p · · · u * 0 and v * = v * q · · · v * 0 . For these dual words we get u * i = v * i for i < k and v * k−1 = u * k−1 ∈ L + . In this case the preceding proof yields an 2-topological space X and a * -morphism f : L → L(X) such thatf (u) * =f (u * ) =f (v * ) =f (v) * , which implies thatf (u) =f (v).
Finally, consider the case: 4) u, v ∈ K L \ K L . Then u = cu ′ and v = cv ′ for some distinct words u ′ , v ′ ∈ K L . By the case (3), we can find a 2-topological space X and a * -morphism f : L → L(X) such thatf (u ′ ) =f (v ′ ). This means that which means thatf (u) =f (v).
The following corollary of Theorem 8.1 implies Theorem 1.2 and shows that the upper bound in Theorem 1.1 is exact.
Proof. Let K 2 L \ ∆ = {(u, v) ∈ K L × K L : u = v}. By Theorem 8.1, for any distinct words u, v ∈ K L there exist a 2-topological space X u,v = (X u,v , (τ ′ u,v , τ ′′ u,v )) and a * -morphism f u,v : L → L(X) whose Kuratowski extensionf u,v : F S L∪{c} → K 2 (X u,v ) separates the words u, v in the sense thatf (u) =f (v). This means that f u,v (u)(A u,v ) =f u,v (v)(A u,v ) for some subset A u,v ⊂ X u,v . Let δ u,v denote the discrete topology on the set X u,v .
Define the L − -topology τ u,v : L − → Top(X u,v ) on X u,v by the formula Consider the L − -topological space X − u,v = (X u,v , τ u,v ) and observe that its full Kuratowski monoid coincides with the full Kuratowski monoid of the 2-topological space X u,v . Moreover, the Kuratowski extensionf u,v : We lose no generality assuming that for any distinct pairs (u, v), (u ′ , v ′ ) ∈ K 2 L \ ∆ the sets X u,v and X u ′ ,v ′ are disjoint. This allows us to consider the disjoint union For every ℓ ∈ L − consider the topology τ ℓ on the set X generated by the base {τ u,v (ℓ) : (u, v) ∈ K 2 L \∆}. We claim that the L − -topological space X = (X, τ ) (which is the direct sum of L − -topological spaces X − u,v ) and the subset A ⊂ X have the desired property: for any two distinct words u, v ∈ K L we getû(A) =v(A), whereû andv are the images of u and v under the (unique) semigroup homomorphism π : F S L∪{c} ) → K 2 (X) such that π(1) = 1 X , π(c) = c X , π(ℓ) =τ ℓ , π(ℓ * ) =τ ℓ for ℓ ∈ L − . This follows from the fact thatû(A) This means that the restriction π : K L → K 2 (X) is injective. By Theorem 7.1, π(K L ) = K(X). Since K 2 (X) = K(X) ∪ {c X • w : w ∈ K(X)}, we get also that π(K L ) = K 2 (X). This means that the homomorphism π maps bijectively the set K L onto K(X) and the set K L onto K 2 (X).

Free Kuratowski monoids
In this section we shall discuss free Kuratowski monoids over pointed linearly ordered sets. By a pointed linearly ordered set we understand a linearly ordered set (L, ≤) with a distinguished point 1 ∈ L called the unit of L. The subsets L − = {x ∈ L : x < 1} and L + = {x ∈ L : 1 < x} are called the negative and positive parts of L, respectively. A function f : L → Λ between two pointed linearly ordered sets is called a morphism if f (1) = 1 and f is monotone in the sense that f (x) ≤ f (y) for any elements x ≤ y of L.
Each pointed linearly ordered set L determines the free Kuratowski monoid F K L defined as follows. On the free semigroup F S L = ∞ n=1 L n consider the smallest compatible partial preorder extending the linear order ≤ of the set L = L 1 ⊂ F S L and containing the pairs (x, x1), (x, 1x), (x1, x), (1x, x), (x, x 2 ), and (x 2 , x) for x ∈ L. The compatible partial preorder generates the congruence ρ = {(v, w) ∈ F S L : v w, w v} on F S L identifying the words x, x1, 1x, x 2 for any x ∈ L. The quotient semigroup F K L = F S L /ρ endowed with the quotient partial order is called the free Kuratowski monoid generated by the pointed linearly ordered set L. By q L : F S L → F K L we shall denote the (monotone) quotient homomorphism. The restriction η L = q L |L : L → F K L is called the canonical embedding of the pointed linearly ordered set L into its free Kuratowski monoid. In Proposition 9.2 we shall see that η is indeed injective.
First we show that the free Kuratowski monoid F K L is free in the categorial sense.
Proposition 9.1. For any pointed linearly ordered set L and any Kuratowski monoid K with a linear generating set Λ any morphism f : L → Λ determines a unique monotone semigroup homomorphismf : Proof. Letf : F S L → K be the unique semigroup homomorphism extending the morphism f : L → Λ ⊂ K.
The partial order ≤ of the Kuratowski monoid K induces the compatible partial preorder on F S L defined by u v ifff (u) ≤f (v). The monotonicity of f implies that the partial preorder contains the linear order of the set L. Then the minimality of the partial preorder implies that ⊂ . This allows us to find a unique monotone semigroup homomorphismf : F K L → K such thatf • q L =f and hencef • η L =f • q L |L =f |L = f .
Proposition 9.2. For any pointed linearly ordered set L the quotient homomorphism q L : F S L → F K L maps bijectively the set K L of Kuratowski words onto F K L .
Proof. Theorem 7.1 implies that q L (K L ) = F K L . To show that q L |K L is injective, we shall apply Theorem 8.2. Choose any injective morphism e : L → L * of the pointed linearly ordered set L into a * -linearly ordered set L * . Letẽ : F S L → F S L * be the unique semigroup homomorphism extending the map e. The injectivity of e implies the injectivity of the homomorphismẽ. By Proposition 9.1, the morphism e determines a unique monotone semigroup homomorphismē : F K L → F K L * such thatē • η L = η L * • e. By Theorem 8.2, there exists a polytopological space X and a * -morphism f : L * → L(X) whose Kuratowski extensionf : F S L * → K(X) maps bijectively the set K L * onto K(X). By Proposition 9.1, the * -morphism f : L * → L(X) ⊂ K(X) determines a (unique) monotone semigroup homomorphismf : F S L * → K(X) such thatf • η L * = f . Thus we obtain the commutative diagram in which the mapf •ẽ|K L is injective. Sincef •ẽ =f •ē • q L , the injectivity of the mapf •ẽ|K L implies the injectivity of the map q L |K L . Since q L (K L ) = F K L , the restriction q L |K L : K L → F K L is bijective.
Now we prove that the congruence ρ on F S L determining the free Kuratowski monoid can be equivalently defined in a more algebraic fashion.
Proposition 9.3. For any pointed linearly ordered set L the congruence ρ on the free semigroup F S L coincides with the smallest congruence ρ on F S L containing the pairs (x, x1), (x, 1x), and (x, x 2 ) for every x ∈ L and the pairs (x 1 y 1 x 2 y 2 , x 1 y 2 ), (y 2 x 2 y 1 x 1 , y 2 x 1 ) for any points x 1 , x 2 , y 1 , y 2 ∈ L with x 1 ≤ x 2 ≤ 1 ≤ y 1 ≤ y 2 .
Denote by ρ ♮ : F S L → F S L /ρ and q L : F S L → F S L /ρ = F K L the quotient homomorphisms. Since ρ ⊂ ρ , there is a unique homomorphism h : F S L /ρ → F K L making the following diagram commutative: In this diagram by i : K L → F S L we denote the identity inclusion of the set K L of Kuratowski words into the free semigroup F S L . The proof of Theorem 7.1 implies that ρ ♮ (K L ) = F S L /ρ.
On the other hand, Proposition 9.2 guarantees that the restriction q L |K L : K L → F K L is bijective. This implies that the homomorphism h is bijective and hence ρ = ρ.
The bijectivity of the restriction q L |K L : K L → F K L and Theorem 5.2 imply: Given two non-negative natural numbers n, p, fix any pointed linearly ordered set L n,p with |(L n,p ) − | = n and |(L n,p ) + | = p and denote the free Kuratowski monoid F K Ln,p by F K n,p .
Proposition 9.5. For any pointed linearly ordered set L and any distinct elements x, y ∈ F K L there is a morphism of pointed linearly ordered sets f : L → L 2,2 such thatf (x) =f (y). This implies that F K L embeds into some power of the free Kuratowski monoid F K 2,2 .
Proof. Enlarge the pointed linearly ordered set L to a * -linearly ordered set L * and denote by e : L → L * the identity embedding. Letẽ : F S L → F S L * be the unique semigroup homomorphism extending e. It is clear thatẽ is an injective map.
By Proposition 9.2, the restriction q L |K L : K L → F K L is bijective. So, we can find Kuratowski words u, v ∈ K L such that q L (u) = x and q L (v) = y. By Theorem 8.1, for the Kuratowski words u, v ∈ K L ⊂ K L * there exist a 2-topological space X and a * -morphism g : L * → L(X) such thatĝ(u) =ĝ(v) whereĝ : F S L * → K(X) is the unique semigroup homomorphism extending the * -morphism g. Moreover, the proof of Theorem 8.1 guarantees that the linear generating set L(X) of the 2-topological space X is isomorphic to the * -linearly ordered set L 2,2 . So, there exists a (unique) bijective * -morphism ι : L 2,2 → L(X). Let f = ι −1 • g : L * → L 2,2 . The commutativity of the following diagram In fact, Proposition 9.5 can be improved as follows.
Proposition 9.6. For any pointed linearly ordered set L and any distinct elements x, y ∈ F K L there is a pair (n, p) ∈ {(1, 2), (2, 1)} and a morphism of pointed linearly ordered sets f : L → L n,p such thatf (x) =f (y). This implies that F K L embeds into some power of the partially ordered monoid F K 1,2 × F K 2,1 .