TOPOLOGIES FOR FINITE WORDS: COMPATIBILITY WITH THE CANTOR TOPOLOGY

. Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of ω -languages it seems to be providing to include finite and infinite words into one (topological) space. In most cases this results in a poor topological structure induced on the subspace of finite words. In the present paper we investigate the possibility to link topologies in the space of finite words with a topology in the space of infinite words via a natural mapping. A requirement in this linking of topologies consists in the compatibility of the topological properties (openness, closedness etc. ) of images with preimages and vice versa. Here, we show that choosing for infinite words the natural topology of the Cantor space and the δ -limit as linking mapping there are several natural topologies on the space of finite words compatible with the topology of the Cantor space. It is interesting to observe that besides the well-known prefix topology there are at least two more whose origin is from language theory – centers and supercenters of languages. We show that several of these topologies on the space of finite words fit into a class of L -topologies and exhibit their special properties w.r.t. to the compatibility with the Cantor topology.


Introduction
Topological methods are useful in the theory of ω-languages in connection with proving hierarchy results (e.g.[6,15,18]).To this end one considers, for a finite alphabet X, the set of all infinite words (ω-words) over X as the infinite product space of the discrete space X.This topology is also known as the Cantor topology.Infinite words may also be viewed as limits of (infinite) increasing families of finite words w.r.t. the prefix ordering.Thus it seems to be providing to include both into the same space.One attempt into this direction was done by Boasson and Nivat [1].Redziejowski [11] observed that the limit considered in [1] is different from that one used in the theory of ω-automata.Therefore he proposed another topology including finite and infinite words into one space.Recently, in [3] possibilities to extend right topologies generated by partial orders on X * to topologies on X ω or X * ∪ X ω were investigated.
Each of these concepts seems to have several drawbacks when considering topology in connection with acceptance of ω-words; the first approach yields a trivial topological structure for finite words whereas the other ones for infinite words give topologies others than the topology of the Cantor space: The topology considered by Boasson and Nivat is closely related to the product or Cantor topology of X ω ; on the one hand its restriction to X ω is the Cantor topology and, on the other hand the whole space X * ∪ X ω is homeomorphic to a closed subset of the Cantor space (X ∪ {⊥}) ω where ⊥ / ∈ X is a new letter.However, all finite word languages are open in this topology, in particular, each finite word is an isolated point in this topology.
Redziejowski's topology has all sets consisting of only infinite words (ω-languages) as closed sets, thus providing no information on the complexity of acceptance by topological means.
Finally, the results of [3] show that unless one uses the prefix order ⊑ the limit process extending the topology on X * yields ambiguous results, more precisely, the limit set may contain more than one element and the topology induced on the space of infinite words may not coincide with the Cantor topology.
In this paper we investigate the possibility to link topologies on the countable set X * to the standard topology [6,15,18] of the Cantor space X ω via a mapping preserving closedness and openness of sets.
As a mapping we use the δ-limit introduced in [4].Because of the countability of X * we cannot expect to obtain a full topological correspondence between the spaces X * and X ω via any mapping.The δ-limit does not go beyond the class G δ of the Borel hierarchy.For the prefix topology, it provides a correspondence between the open and closed sets in X * and the Cantor space, respectively, and furthermore, of the (σ, δ)-subsets of X * as introduced in [14] and subsets of X ω being simultaneously of type F σ and G δ (see [15], Sect.2.4).
Among the topologies on X * we consider here are the topologies induced by the Anf L -operators introduced in [7][8][9].Particular cases of these topologies are the topologies induced by the centers [1,2,9] and supercenters [17] of languages.It turns out that both topologies play a special rôle: the center-topology being the coarsest one having all finite sets closed, and the supercenter-topology being the finest one for which the prefix set pref (F ) of every closed subset F ⊆ X ω is the smallest closed set in X * corresponding to F via the δ-limit.
The paper is organised as follows.After the notation used we introduce some topological background emphasising properties of the derived set operator.Then we deal with several topologies for finite and infinite words.Section 4 describes the linking of topologies for finite words with the Cantor topology for ω-words.A fundamental property for a topology of finite words is the compatibility with the topology of infinite words.The class of L-topologies introduced by Prodinger and Urbanek [7,9] is the topic of Section 5.In the subsequent part we deal with topologies fulfilling a strengthened compatibility condition.Some of these results were announced in [16].Finally we show some limitations of L-topologies in respect to compatibility with the Cantor topology.

Notation
We introduce the notation used throughout the paper.By N = {0, 1, 2, . ..} we denote the set of natural numbers.Let X be an alphabet of cardinality |X| ≥ 2, a, b ∈ X, a ̸ = b.By X * we denote the set (monoid) of words on X, including the empty word e, and X ω is the set of infinite sequences (ω-words) over X.For w ∈ X * and η ∈ X * ∪ X ω let w • η be their concatenation.This concatenation product extends in an obvious way to subsets W ⊆ X * and B ⊆ X * ∪ X ω .For a language W let W * := i∈N W i be the submonoid of X * generated by W , and by we denote the set of infinite strings formed by concatenating words in W . Furthermore |w| is the length of the word w ∈ X * and pref (B) is the set of all finite prefixes of strings in B ⊆ X * ∪ X ω .We shall abbreviate w ∈ pref (η) (η ∈ X * ∪ X ω ) by w ⊑ η.
Further we denote by B/w := {η : w • η ∈ B} the left derivative (left quotient) of the set B ⊆ X * ∪ X ω generated by the word w.
In the theory of ω-automata, ω-words are introduced as upper bounds of infinite chains of words ordered by the prefix relation "⊏"; this is reflected by the following limit operation (see [3,4,11,14,15]).
The δ-limit of a language W ⊆ X * is defined as

General topology
Usually, a topology T = (X , O) on a set (space) X is given by a family of open sets O ⊆ 2 X .Here O is a family of subsets of X containing X and closed under arbitrary (including empty) union and finite intersection.
Following Kuratowski (cf.[5,10]) one can also define a topology via a closure operator.A mapping α : 2 X → 2 X is called a topological closure operator provided it satisfies the following conditions. ) In view of equation (2.3) the identity in equation (2.5) can be replaced by α(α(M )) ⊆ α(M ).
By T α we denote the topology open.This is equivalent to the fact that it holds where α i , i = 1, 2 is the closure generating the topology T i .
A closure operator may also be obtained via the Cantor-Bendixson derived set (set of accumulation points) M d of M ⊆ X [5], Chapter 1, Section 9, III.
The derived set operator d has to satisfy the following conditions.
In particular, every closure α is also a derived set operator.
It is readily seen that α d (M ) := M ∪ M d is a topological closure operator.In view of equation (2.9) M d is closed in T α d , and more generally, a set M is closed in T α d if and only if M d ⊆ M .Via α d , a derived set operator d defines a topology on X .
Property 2.1.Every M ′ , where Property 2.2.Let γ : 2 X → 2 X be a derived set or closure operator.Then if G is open in the topology defined by γ.
For the sake of completeness we add a proof.
Proof.The inclusion ⊇ is obvious.From equation (2.8) we have γ

Topologies for Words and ω-words
In this section we present some known topologies for finite and infinite words.

The Cantor topology on X ω
The first one is the widely investigated Cantor topology.We consider the set X ω as a metric space (Cantor space) (X ω , ρ) of all ω-words over the alphabet X where the metric ρ is defined as follows.
Since X is finite, this space is compact, and C(F The topology can be defined by the metric ρ as in equation (3.1) or, alternatively, by letting As (X ω , ρ) is a metric space the Cantor-Bendixson derived set F d of a subset F ⊆ X ω can be described as We conclude this part with a relation between the δ-limit and G δ -sets in Cantor space.

Proposition 3.1 ([4]
).A subset F ⊆ X ω is a G δ -set if and only if there is a language W ⊆ X * such that F = W δ .

The prefix topology T p on X *
A well-known topology on X * which resembles the product or Cantor topology is the prefix topology T p on X * , that is, the right topology on X * derived from the prefix order ⊑ on X * .
The open subsets in T p are of the form W • X * where W ⊆ X * .As it is easily verified the closure operator defining this topology is the initial word operator pref assigning to each language W ⊆ X * its set of prefixes pref (W ) : It has the following property.
Lemma 3.2.The prefix topology T p is the coarsest topology having all subsets pref (F ) ⊆ X * , F ⊆ X ω , closed.
Proof.This follows from the identity pref In this topology, however, not every finite set W ⊆ X * is closed.It holds only the so-called T 0 -condition: For every pair w, v ∈ X * , w ̸ = v, there is an open set W ⊆ X * which contains exactly one of the words w or v.

Topologies on X * ∪ X ω
In this section we consider several topologies on the space of finite and infinite words X * ∪ X ω and its restrictions to the sets X ω and X * , respectively.
The first one resembles the Scott topology on X * ∪ X ω (cf.[13]).Its open sets are of the form W • (X * ∪ X ω ).The restriction to X ω is the Cantor topology and the restriction to X * is the prefix topology T p .
The topology considered by Boasson and Nivat [1] is closely related to the product or Cantor topology of X ω ; on the one hand its restriction to X ω is the Cantor topology and, on the other hand the whole space X * ∪ X ω is a homeomorphic to a closed subset of the Cantor space (X ∪ {⊥}) ω where ⊥ / ∈ X is a new letter.However, all finite word languages are open in this topology, in particular, each finite word is an isolated point in this topology.Thus its restriction to X * is the (trivial) discrete topology.
Redziejowski [11] observed that the limit considered in [1] is different from the one used in the theory of ω-automata.Therefore he proposed another topology on the space of finite and infinite words.Here the closure of a subset W ∪ F ⊆ X * ∪ X ω , W ⊆ X * , F ⊆ X ω , can be described via the δ-limit as (cf.[11], Property 4.4 (1)) Then subsets of the form F ⊆ X ω and W ∪ X ω , W ⊆ X * , are closed, and, consequently, the restrictions to X ω and X * , respectively, are the (trivial) discrete topologies.

Linking topologies on X * to Cantor topology
The open sets in T p resemble those in Cantor space.One easily observes, that due to the identity (W 2) and Lemma 3.2 there is a close correspondence between open (closed) subsets in the prefix topology on X * and Cantor topology, respectively.

Compatibility of topologies
The connection between the prefix topology on X * and the Cantor topology on X ω via the δ-limit is shown in [15], Section 2.4.This connection fulfils the following property.
Therefore we consider topologies on X * which are linked via our δ-limit to the Cantor topology on X ω .Then every language W ⊆ X * has as its image the G δ -set W δ ⊆ X ω .(In fact, because of the different cardinalities of the spaces X * and X ω , we cannot expect to obtain every subset of X ω as an image.)Definition 4.1 requires that the image of every open (closed) language W ⊆ X * is also open (closed), and every open (closed) ω-language F ⊆ X ω is the image of an appropriately chosen open (closed) language.Lemma 4.2.Let a topology T α be compatible with the Cantor topology.Then the closure α : 2 X * → 2 X * satisfies the following inclusions The second one follows from the fact that α(W ) δ is a closed set containing W δ .
Moreover, Conditions 1 and 2 of Definition 4.1 are inherited in the following way.

If the topology
Thus, if the topologies T 1 , T 2 are compatible with the Cantor topology then every topology T which is finer than T 1 and coarser than T 2 is also compatible with the Cantor topology.This is due to the identity (W 2) and Lemma 3.2 as it was mentioned above.

(σ, δ)-subsets of X *
As Proposition 3.1 shows, the δ-limit can map languages only to ω-languages in the Borel-class G δ , it is interesting to observe that we can also characterise those G δ -sets which are simultaneously F σ -sets via their δ-preimages (see [14]).These subsets of X * show also some special properties w.r.t.topologies on X * .
We start with some easily verified properties of the δ-limit defined in equation (2.1).
The identity in equation (4.4) follows from applying equation Then we have the announced connection to F σ -sets in Cantor space.
Examples of (σ, δ)-subsets of X * are languages of the form pref (W ), W • X * , and W ⊆ X * such that W δ = ∅.Definition 4.5 is equivalent to the following one We have also the following equivalent property.
Proof.We show first that 1. is equivalent to 2. The inclusion Let W be a (σ, δ)-subset of X * , and let As, by definition, the complement of a (σ, δ)-subset is also a (σ, δ)-subset, replacing W by X * \ W shows the equivalence of Items 1 and 3.
Moreover, the following is true.
We finish this section with a sufficient condition for the compatibility of topologies T = (X * , O) which is immediate from equation (4.5).

L-topologies
In this section we investigate a method for defining a class of topologies on X * which are finer that the prefix topology T p .These so-called L-topologies were introduced by Prodinger [7,8] using the Anf L -operator (see also [9]).Under several conditions this Anf L -operator is a derived set operator on X * , thus defines a topologythe L-topology on X * .The corresponding closure will be denoted by α L , that is,

The operator Anf
In [7,9] Prodinger and Urbanek defined a generalisation of the initial word operator pref as follows.Let L ⊆ 2 X * be a family of languages, W ⊆ X * , and define The operator Anf L has the following properties (see [7,9,12]): Property 5.2.The following conditions are equivalent.
Next we give some relations to the prefix-operator pref .
We continue with the requirement under which the operator Anf L satisfies the condition of equation (2.9) of a derived set operator.

L-topologies
Here we investigate under which conditions an Anf L -operator has the properties of a derived set operator on X * (cf.Prop.5.1, 5.2 and Lem.5.4).The following theorem is an analogue to Theorem 2.3 of [7] for derived set operators.
Theorem 5.5.A mapping Anf L is a derived set operator on X * if and only if the following conditions are satisfied.

∅ /
∈ L, 2. W ∈ L and W ⊆ V imply V ∈ L, and W ∪ V ∈ L implies W ∈ L or V ∈ L, and 3. Anf L (W ) ∈ L implies W ∈ L.
2. Theorem 4.13 of [9] shows that this is equivalent to 3. Lemma 5.4 shows that Item 3 is equivalent to equation (2.9).
Simple examples of L-topologies are the prefix topology T p and the discrete topology D = (X * , 2 X * ).Here one can choose L p = {W : W ̸ = ∅}, and L D = ∅, respectively.This yields Anf Lp = pref and Anf L D (W ) = ∅, for all W ⊆ X * .
The following shows a sufficient condition for Item 3 of Theorem 5.5.
Proof.Let Anf L (W ) ∈ L. Then Anf L (W ) ̸ = ∅.Now in view of the equivalence 1 ↔ 2 of Proposition 5.3 we obtain W ∈ L.
Next, we give an example that the condition ∀W ∀w(W/w ∈ L → W ∈ L) is not necessary.
Example 5.7.Define L := {V : V ∩ (X 2 ) * ̸ = ⊘}.Then a • (X 2 ) * / ∈ L. Conditions 1 and 2 of Theorem 5.5 are trivially satisfied.To verify Condition 3 we observe that, if Anf L W ∈ L it contains a word v of even length.Then W/v ∈ L. Thus W/v contains also a word w of even length, whence wv ∈ W , that is, W ∈ L.
Since every L-topology is finer than the prefix topology T p , Lemma 4.3.2 and Corollary 4.4 yield the following.Proposition 5.8.An L-topology is compatible with the Cantor topology on X ω if and only if for every W ⊆ X * , the set α L (W ) δ is closed and the set (X * \ α L (W )) δ is open in Cantor space.
Finally, we mention that L-spaces can be characterised also via their open sets.Theorem 5.9 ([7], Thm.2.16).A topological space T = (X * , O) is an L-space if and only if W ∈ O implies w • W, W/w ∈ O for w ∈ X * .

Strongly compatible topologies on X *
In this section we consider topologies on X * compatible with the Cantor topology.Several of them can be defined using the apparatus introduced in the preceding section.

Center and supercenter topologies
Special Anf L -operators were considered in connection with language-theoretic questions.These were referred to as centers [2] or supercenters [17], respectively, of languages.Definition 6.1 (Center).
Since the alphabet X is finite, König's infinity lemma shows that W ⊆ X * is infinite if and only if pref (W ) is infinite which is equivalent to pref (W ) δ ̸ = ∅.Thus we have center(W ) = sctr(pref (W )) = pref (pref (W ) δ ) . (6.1) Both families L c and L sc satisfy the conditions of Theorem 5.5 and Proposition 5.6.So center and sctr are derived set operators and define topologies T c and T sc , respectively, on X * .As L-topologies the center or supercenter topologies are finer than the prefix topology T p .Moreover, equation (5.2) shows that T sc is finer than T c . In . This shows that T sc does not coincide with T c .
The prefix topology has the property that not every finite subset of X * is closed.The center topology proves to be the coarsest topology refining T p and having all finite sets closed.To this end we show the following.Theorem 6.3.If α : 2 X * → 2 X * is a topological closure such that all pref (F ), F ⊆ X ω , and all finite sets are closed then Proof.If all pref (F ), F ⊆ X ω , are α-closed, then according to Lemma 3.2 all pref (W ), W ⊆ X * , are α-closed, whence α(W ) ⊆ pref (W ); moreover, all u • X * are α-open.
We show α(W Corollary 6.4.The topology T c is the coarsest topology having all subsets pref (F ) ⊆ X * and all finite subsets closed.
The next lemma shows a connection between the supercenter topology and (σ, δ)-subsets of X * .Lemma 6.5.If a topology T on X * is coarser than the supercenter topology T sc then every open or closed set is a (σ, δ)-subset X * . Proof.
The proof of Lemma 6.5 shows also that every set open in the supercenter topology has the form W • X * \ V where V δ = ∅.
Then the identity (W in connection with Lemma 4.9 shows that T sc is compatible with the Cantor topology.

Strong compatibility -characterisation
In every topology finer than the prefix topology T p all languages pref (F ) where F ⊆ X ω are closed.Moreover, pref (F ) δ = C(F ).In this section we are investigating which topologies have the languages pref (F ) as smallest closed sets V yielding V δ = C(F ).It turns out that supercenters of languages play a crucial rôle in this respect.
In particular, every pref (F ), F ⊆ X ω , is closed in T .
Theorem 6.7.A topology T on X * is strongly compatible if and only if the corresponding closure operator α T satisfies Proof.If T is strongly compatible then Lemma 3.2 shows that T refines the prefix topology T p .Hence α T (W ) ⊆ pref (W ); and equation (6.2), for To prove the converse, we refer to equations (3.2) and (6.3).This shows pref In view of equation (2.6) the inequality of Theorem 6.7 shows that a topology T on X * is strongly compatible if and only if T is finer than T p and coarser than T sc .Then, since both topologies T p and T sc are compatible with the Cantor topology, Lemma 4.3 yields the following.Corollary 6.8.Every strongly compatible topology on X * is compatible with the Cantor topology.

Strong compatibility and L-topologies
The so far considered strongly compatible topologies T p , T c and T sc are L-topologies.It arises the question whether all strongly compatible topologies are L-topologies.In view of Theorem 6.7 this is equivalent to whether all topologies between the prefix and the supercenter topology are L-topologies.
A further observation is that as a consequence of Lemma 6.5 and Theorem 6.7, the closed and the open sets in every topology strongly compatible with the Cantor topology are always (σ, δ)-subsets of X * .We also address the question whether all compatible L-topologies having as open sets (σ, δ)-subsets of X * are strongly compatible.
In this section we will show that for both instances we find counter-examples.To this end we use the well-known possibility to define topologies via their bases (e.g.[10], Ch.I, Sects.2, 2.1): Property 6.9.Let X be a set and B ⊆ X be closed under intersection.Then for O := { M ∈A M : A ⊆ B} the pair T = (X , O) is a topological space with open sets O. Remark 6.10.The set B in Property 6.9 has the properties of a base of the topological space T , as it generates all its open sets.In topology, however, it is not required that a base be closed under intersection [5,10].
The first example is a topology on X * which is strongly compatible with the Cantor topology but not an L-topology.Since the topology T = (X * , O) of Example 6.11 is coarser than T sc , its open sets are (σ, δ)-subsets of X * .Next we provide an example of a compatible but not strongly compatible L-topology having all open sets as (σ, δ)-subsets of X * .Lemma 6.12.There are compatible L-topologies on X * such that all its open sets are (σ, δ)-subsets of X * which are not strongly compatible with the Cantor topology.
Proof.We construct an L-topology T = (X * , O) such that every open set W ∈ O and hence also every closed set is a (σ, δ)-subset.To this end we use Theorem 5.9 and Property 6.9.
We let X = {a, b} and B = w Then every open set has the form3 First we show that O is closed under the operations w • and /w.Closure under premultiplication with a word is trivial.It remains to show that (w In the former case (w Next, we show that every open set W ⊆ X * is a (σ, δ)-subset of X * , that is, for ξ ∈ W δ we have to show that pref (ξ) \ W is finite.If ξ ∈ W δ in view of equation (6.4) there is an For the compatibility of T = (X * , O) with the Cantor topology, in view of Lemma 4.9, it suffices to show that W δ is open if W is given by equation (6.4).First observe that W δ ⊆ i∈N w i • X ω and w ) is not strongly compatible with the Cantor topology.

Miscellaneous
In this section we first present examples of topologies which are compatible with the Cantor topology but not strongly compatible.Then we consider L-topologies related to the Cantor-Bendixson derived set in Cantor space.

Some examples
The first one is coarser than the prefix topology, and the second one is a refinement of the first one incomparable with the prefix topology and the center topology.For both topologies the open (and closed) sets are (σ, δ)-subsets.
∅} and define O 2sc as in Property 6.9.Then B 2sc is closed under finite intersection and O 2sc consists solely of (σ, δ)-subsets of X * .
Proof.Closure under intersection follows from the identities (W Finally Thus the topology T 2sc is not finer than the prefix topology T p .Since every U with U δ = ∅ is closed in T 2sc , in particular, every finite set is closed in T 2sc , the topology is not coarser than T c .

Cantor-Bendixson-topology
In this section we first present two examples of L-topologies which are strictly finer than the supercenter topology.Thus they are not strongly compatible.The first example is related to the Cantor-Bendixson derived set in Cantor space and compatible with the Cantor topology.The second one refers to the Cantor-Bendixson Theorem as it is concerned with condensation points in W δ , that is, points ξ ∈ X ω for which every w • X ω ∩ W δ , w ⊏ ξ, is uncountable.Theorem 7.4.Let L ∞ := {W : W δ is infinite }.Then Anf L∞ is a derived set operator on X * and the topology defined by Anf L∞ is compatible with the Cantor topology.
Proof.It is readily seen that Anf L∞ satisfies Conditions 1 and 2 of Theorem 5.5 and the hypotheses of Proposition 5.6.Thus Anf L∞ is a derived set operator which, according to Proposition 5.3, is prefix closed.Now, in view of Proposition 5.8 it suffices to show that, for the closure α L∞ of the topology generated by Anf L∞ , the ω-languages α L∞ (W ) δ are closed, and (X * \ α L∞ (W )) δ are open in Cantor space.
Observe that w ∈ Anf L∞ (W ) is equivalent to W δ /w is infinite which in turn is equivalent to W δ ∩ w • X ω is infinite.Thus, ξ ∈ Anf L∞ (W ) δ if and only if W δ ∩ w • X ω is infinite for all w ∈ pref (ξ), that is, in view of equation (3.3) Anf L∞ (W ) δ is the Cantor-Bendixson derived set of W δ .This shows that α L∞ (W ) δ = W δ ∪ Anf L∞ (W ) δ = C(W δ ) is closed.
It remains to prove that (X ω \ α L∞ (W )) δ is open.To this end we use again the fact that Anf L∞ (W ) δ is the Cantor-Bendixson derived set of W δ and Anf L∞ (W ) = pref (Anf L∞ (W )) is a (σ, δ)-subset of X * .
The second assertion is obvious.
Finally, we give a non-trivial example4 of an L-topology not compatible with the Cantor topology.
and we have γ(M ) ⊆ γ(M ∩ G) ∪ (X \ G) whence the reverse inclusion follows via intersection with the set G.As usual countable unions of closed sets are called F σ -sets and countable intersections of open sets are called G δ -sets.
Lemma 4.3.2 has the following special instance.Corollary 4.4.If a topology T on X * is finer than the prefix topology T p then every closed subset F ⊆ X ω has the closed δ-preimage pref (F ), and every open subset W • X ω has the open δ-preimage W • X * .

Lemma 4 . 9 .
Let T = (X * , O) be a topology on X * such that every open set W ∈ O is a (σ, δ)-subset of X * .Then T is compatible with the Cantor topology if W δ is open for every W ∈ O and every open subset E ⊆ X ω has an open δ-preimage V ∈ O.

Example 6 . 11 .
Let X = {a, b} and define B := {w • X * \ V : w ∈ X * ∧ V ⊆ a * • b} and the family of open sets O as in Property 6.9.Then T = (X * , O) is a topology finer than the prefix topology T p and, since V δ = ∅ for V ⊆ a * • b, coarser than T sc .Consider W := b • (X * \ a * • b).Then b ∈ W and bb / ∈ W .The smallest open set containing the word b is b • X * \ a • a * • b which contains the word bb.Thus W is not open, and according to Theorem 5.9 the topology T is no L-topology.

Example 7 . 1 .
Define the topology T 2p by O