2) TOPOLOGICAL LINGUISTICS AT A FIRST GLANCE

Thus far we have characterized topological linguistics in a rather informal way. This starting point has been adopted in order to attain an intuitive confidence in the topological nature of the epistemological problem that all linguists must deal with. But the time has come to go a step further into the notion of topological space. In fact, the present proposal -conceived as a conciliation of alternative doctrines, rather than as a new one- will prove to be a suitable one as it takes advantage of the possibilities that topology is able to provide. 
Formalization, although it will be kept to a minimum, can no longer be delayed. How could we imagine the topological space that linguistic elements and their possible describing subsets are said to constitute? The first problem arises when considering the nature of the elements described in metalinguistic statements: are they entire utterances, words, or some other kind of minor constituents (i.e. morphemes or phonemes)? If we observe the nature of the metalinguistic statements which are usually found in grammars, we discover that there is not one all-embracing answer: linguistic elements from the metalinguistic point of view are either phonemes -"[b] (is a bilabial stop)", or syllables -"[bei] (is the first syllable of baby )", or words -"['beibi] (is the subject of the baby looked at her mother )", or even full utterances -/'beibis a:r 'nais/ (is an attributive sentence)", etc. However, metalinguistic elements seem to be neither phonemes or syllables, nor morphemes or words, but they  are always sentences -"[b] is a bilabial stop", etc.
Let us examine the formal model we are introducing in this chapter in order to obtain some pointers that could help us to solve the question raised above. The definition of topological space would read as follows :
A topology of open sets is said to be defined in a set X when there exists in X a system T of subsets of X, the T-sets, for which the following axioms are verified : 
a)The finite intersection of open sets of T belongs to T.
b)The infinite sum of any number of open sets of T belongs to T.
c)The set X and the empty set ø are open sets, and they are closed sets at the same time.

Although topological spaces can be found in many situations, human languages and the grammatical metalanguages that intend to describe them fulfill their formal requirements quite well. Let X be the set of all linguistic elements that constitute a human language -i.e X=L-, and T the system of subsets that are used to describe them, that is their grammar -i.e. T=M. By doing so we could interpret our above axioms as follows:
a) The intersection of metalinguistic sets belongs to metalanguage too. Such a formulation has allowed grammarians to avoid the use of metalinguistic items when they lacked internal adequacy requirements. In other words: how could we guarantee that verb  is an adequate term when speaking of English grammar, but that aorist  -as found in Greek or in Russian- or any other capricious terms are not? The answer to this question would be that verb  is precisely the feature that linguistic elements of a natural class of English language have in common and that, consequently, it appears in all their metalinguistic descriptions and remains as the result of their intersection ("to go" is an intransitive verb, "to eat" is a transitive verb, the verb  of "the boy comes to London" agrees with its subject, etc); on the other hand, aorist  could not properly be used to describe any class of English linguistic elements, though Greek or Russian utterances which contain aorists can be perfectly translated  into English by means of adequate sentences with verbal forms that could individually be called aoristic  (but not aorists ). Condition a) exemplifies the convenience of using distributional classes whose logical intensions constitute the grammatical terms, in linguistics.
But one must bear in mind that this intersection of metalinguistic utterances must be finite: if we could intersect all the statements that grammar can, then we would obtain at least one feature they share (their intersection not being 0), and consequently all linguistic elements could be characterized by this feature, be it "sign" (which excludes the possibility of having phonemes as elements of L), be it "morpheme" (which leaves out, for example, utterances, etc), and so on.
Because the intersection of metalinguistic elements should be a finite process, as stated in point a), we can definitionally guarantee that the set of elements that constitutes the linguistic topological space looks like a heterogeneous set with several levels of analysis necessary to describe it (phonic level, morphic level, word level, phrase level, etc). Notice that a) meets the methodological requirements that any descriptive linguistic approach has to fulfill: hence, topological space constitutes an adequate formal model for the application of that method, or at least a formalism that is as adequate as common set and graph theory procedures have been until now.

b) However, curiously enough, while a) meets the requirements of descriptive linguistics, b) seems to fit the methodological necessities of generative grammar. Whereas the intersection of metalinguistic patterns should not go on infinitely so as to avoid level mixing when describing a corpus, their sum should be carried out infinitely because the goal of grammar is now the exact opposite, namely to mix up all levels as far as the maximal unit of speech act. 
The generation of a sentence cannot be accounted for without adding succesively increasing metalinguistic instructions to a set of linguistic items. For example, the boy saw the girl  is made out of  the (Det), boy ( Noun), see  (Verb), girl  (Noun) as found in a dictionary: but the subject noun phrase the boy  consists of the (Det) + boy  (Noun) + [noun phrase], the latter being purely metalinguistic information, the predicate verb phrase consists of see  (Verb) + the  (Det) + girl  (Noun) + {verbal phrase}, the information in brackets being metalinguistic too, etc. If we take the analysis a step further we find that the sentence consists of the  (Det) + boy  (Noun) + see  (Verb) + the  (Det) + girl (Noun) + {subject+predicate=sentence}, etc. At the same time morphological information -that changes see  into saw  for example- has to have been  merged with syntactic and phonological information as well.
In other words: when generating a sentence, whether we adopt the point of view of the hearer or that of the speaker, all information must be able to be joined together freely -i.e "an infinite sum of metalinguistic elements..."-  and each step of the generation process adds new metalinguistic (and even linguistic) information provided by grammatical rules -i.e "...being metalinguistic too"-: both requirements are included in the above definition.
Generative grammarians disagree on the levels where linguistic and metalinguistic information may be inserted by means of the generation process of a sentence, but all of them accept that modularity is not recoverable, and that consequently the boundaries among levels and components are lost when dealing with real utterances and that ambiguities become holistic rather than syntactic, semantic, or otherwise (N. Chomsky, 1977, Essays on Form and Interpretation, 4). This would be the proper interpretation of the axiom which states that the infinite sum of metalinguistic elements  should be metalinguistic as well. 
 Grammarians usually coincide with W. von Humboldt in stating that grammar implies an infinite use of finite resources. And certainly, that is so: a finite set of lexical items can be combined infinitely by the rules of grammar: this infinite capacity of grammar is expressed in point b) which could be considered an alternative formulation of the recursive property of grammar.

c) Point c), as found in the definition of topological space, meets the epistemological background of any structuralist approach to human language by establishing the "made out of" quality of the metalanguage-language relationship. As we said in chapter 1), the problem that structuralism faces is to demonstrate that its proposed metalinguistic structures are suitable for the linguistic utterances they are supposed to support. But c) clearly states that, although M is a system of subsets of L and is consequently contained in L, at the same time L belongs to M as an element of it, that is M is of an L nature and vice versa. 
The contradictory nature of the L-M relationship has been pointed out by every structuralist school of linguistics. As F. de Saussure noticed seventy years ago, although M (the "langue") must precede L (the "parole") in order to establish it, from the historical point of view L has to come before M. Now we can answer this intriguing question in topological terms because it was, from the beginning, a topological problem that should have been solved topologically too. M precedes L in the sense that it can speak about L which belongs to M (L Œ  M), that is, M has L as part of its alphabet; L includes M in the sense that M is of an L nature  (L … M), whether it is a set of metalinguistic statements of the form "boy is a noun", etc, as found in languages that possess an explicit grammar, or whether it simply consists of a set of significant environments such as #the...comes / leaves / eats#, that constitute the implicit grammar.
Keep in mind that, as a member of the M system of sets, L becomes an open and a closed set at the same time. But, as the description of L by M is generally a division according to significant M patterns, the trigger of such a dividing process, i.e. the zero sign 0, must be linguistic and metalinguistic (i.e. closed and open) simultaneously, as point c) also thoroughly states.
To sum up: topological linguistic space -according to the above definition whose three main points each meet one of the three conflicting theories in linguistics- can be conceived of as a set of linguistic elements of several types (phones, morphs, words, phrases, sentences, etc), where some of them describe the others. Depending on the purpose of the investigation (i.e. on the kind of facts we are interested in), it will be necessary to choose as metalinguistic items some linguistic elements over others, and in each occasion we are said to have introduced a different topology into the space. 
For example, to describe phonemes there is no sense in choosing phrase environments, but it would be advisable to select syllabic ones: thus, in root  vs. rat  we define /u:/ and /æ/ because they alternate in /r...t/ environment, but we should not oppose them in a typical environment such as I found a root/rat in the garden, because other noun environments behave in a different manner -the *root/rat bit me-.