4) FURTHER STEPS IN TOPOLOGY

Topology is accustomed to facing the methodological problem of constructing new topological spaces from old given spaces. One of these methods involves assigning a standard sort of topology to the Cartesian product of spaces, thus building a new space from the originally given ones -product topology. The second method of constructing a new space from a given one is based on dividing the given space X into equivalence classes, each of which is a point of the newly constructed space. In this case we "identify" the points of certain subsets of X obtaining a new set of points which is assigned the quotient topology  .
Roughly speaking we could say that product topologies join some sets of elements, each with their own topology, to new sets whose elements consist of associations of the former elements and whose topologies are constructed by means of the former topologies too. Text grammars provide a good exemplification of such a situation: first we consider a specific language and its grammar (the set of linguistic elements and their metalinguistic patterns), the social implications of linguistic signs along with internal describing keys, the psychological values of linguistic signs along with the psychological variables that are intended to measure them, etc; secondly we join these elements by making up series where a linguistic sign has a social implication and a psychological value etc., and we expect these associations -i.e., the texts - can be explained by a new topology (product topology) that is derived from the former topologies each set of elements had on its own. This theoretical possibility has yet to be explored in liminar grammar because it implies the previous constitution of topological spaces in linguistics, sociology, psychology, etc. (in any case, topological considerations are being increasingly employed in all human sciences, hence a forthcoming topological text grammar can be expected soon).
Quotient topology, on the other hand, seems to be an adequate method for syntax. This is due to the fact that syntax cannot be conceived of without simultaneously considering total-sound, partial-sound, and sound-meaning data, and therefore represents a joining process of some of them. For example, when we define the boy  as the "subject" (a typical syntactic label) of the sentence the boy likes music very much,  we are dealing with:
i) total-sound facts; the position of the boy before the verbal form likes.
ii) partial-sound facts; the lack of [-s] in boy  and the presence of [-s] in likes  that permit us to declare that boy  and likes  agree.
iii) sound-meaning facts; the "animate" character of boy  as opposed to the "inanimate" nature of music  which makes the first more appropriate as a doer of the action than the second. 
Notice the relative character of partial-sound facts -boy  agrees with likes  in this sentence, but it would not agree in the boy/we/you liked it -, as opposed to the absolute nature of either total-sound or sound-meaning facts -boy   must always precede the verb when it is the subject, boy  is always "animate", etc..
In order to put these kinds of facts together we need to construct classes of equivalence among total-sound, partial-sound, and sound-meaning elements. An equivalence relationship R is said to be held among the elements of a set if they are joined in groups of -at least- three elements, such that they present reflexive property (aRa), symmetric property (aRb implies bRa), and transitive property (aRb and bRc imply aRc). It is easy to understand that i, ii, and iii satisfy the above requirements because each relates to itself -reflexivity-, the order of their relationship is not fixed (in fact it changes in the coding and in the decoding processes), and all of them relate to the others in order to constitute a new equivalence class that defines the "subject".
Now we can introduce the rather awkward definition of quotient topology that lies at the foundations of syntax:
"Divide a set L (for example the set of total-sound, partial-sound, and sound-meaning linguistic elements that constitute a human language) into arbitrary subsets that are not empty such that Ci « Ci' = 0 for i ð i' through a relationship of equivalence on the elements of L. Next we define the natural application j that makes its continent set Ci, understood as a new set L' (the set of linguistic signs), correspond to each la  of L. If a topology M (a grammar) is defined on L, beginning with the same topology it is possible to define a quotient topology M' on L' as follows: we say that a subset M' Œ L' is open when the reverse image with respect to j , that is j-1(M'), is open in L".
Although it may appear difficult, this definition simply establishes that syntax as a method relies on phonology, morphology, and semantics because its object of study, the linguistic signs, consists of phonic, morphological, and semantic data, and this fact allows syntax to take advantage of their methodological patterns as well.
 In summary:
elements of L'  elements of L     elements of M            elements of M'
sign boy         t-sound [boi]    1st position (phonol)       [noun,
                     p-sound 'boy'     Nø + Vs   (morphol)        subject
             sound-mean."boy"    "animate"(semant)        (syntax)]

However, a quotient topology is not simply a sum of other topologies but a topology that pivots on them; similarly, a sign does not only stem from adding up phonic, morphologic, and semantic data but it is supposed to be a new whole at the same time. In fact, one of the common shortcomings of early distributionalism and structuralism, that generative grammar intended to solve, was the tendency to end grammatical description when phonic, morphological, and semantic inventories had been established, and to assume that syntax simply had to consist of putting them together.
To define syntax as a quotient topology means that syntactic methods and categories must share some things in common with phonological, morphological, and semantic ones, but also that they are different. There must be, therefore, purely phonic, morphic, and semic descriptions of a language that apply to linguistic space before  an equivalence relationship is introduced in it and, on the other hand, there are syntactic-phonological, syntactic-morphological, and syntactic-semantic descriptions that only appear after  this very relationship has been constituted (the words before  and after  not necessarily having a chronological significance, but perhaps only a causal one).  The former phenomena require their own topologies -induced topologies- and should be named in a specific way (I propose calling them phonematics, morphematics, and semematics respectively):

phonematics   æ  phonic events              ØØ          phono-syntax
morphematics      morphic events        ØØ           morpho-syntax
semematics     æ  semic events               ØØ            semo-syntax
induced topology                                 equivalence  quotient topology
     (before)                                               relationship   (after)                

This raises the question of which would be the more adequate topology for phonematics, morphematics, and semematics. Notice that these methods deal with exclusively  phonic/morphic/semic facts, and for this reason they are of a paradigmatic  nature because linguistic facts, when lineally combined, have also syntactic implications, but when arranged in memory chunks they lack any relationship with syntax (on the contrary, syntactic methods, whether phono-, morpho-, or semo- syntactic, are always syntagmatic). 
On the total-sound axis we can say that the paradigmatic opposition between tin  and pin , which permits us to recognize the phonemes [t] vs. [p], is phonematic ; on the other hand, the syntagmatic associations of consonantal clusters are phonosyntactic  because they display syntactic significance (for example "...tp..." marks a word limit in English, that is, "...t#p...", as in flat part, Pat played it etc.). In a similar way man/men  or boy/boys  are morphematic  allomorphs of the English singular/plural paradigm, whereas this very men  when imposed by the plural form of the verb as in the men come  exhibits morphosyntactical  significance. The semic feature [+male] of boy  as opposed to [+female} of girl  is a semematic  fact, but the semic feature [+animate] which is responsible of boy  being the doer of the boy watered the garden  could be conceived of as a semosyntactic  remark.
In order to account for paradigmatic facts and to provide adequate methodologies (phonematics, morphematics, and semematics) we have to choose a kind of topology that will, starting from the set of linguistic data  before the equivalence relationship has been established (thus phonic, morphic, and semic data are not yet joined), select only those facts which belong to the same axis of description -i.e., either phonic, or morphic, or semic-, and at the same time the metalinguistic descriptions which suit them. Such a topology is called induced topology and its definition reads as follows: 
"Let L' be a fixed subset of a topological space {L,{M}} and Y any open set in {L,{M}}. The system of all the subsets of the form L' « Y that are open in {L,{M}} constitute a topology for L', the induced  topology  M'  or topology relative to the topology M, which constitutes with L' the subspace {L',{M'}} (conventionally any topological space where X is the system of closed sets, and T the subsystem of open ones will be represented {X,{T}})".
And, surely, this is the way paradigmatic procedures proceed. Suppose L' is the set of all phonic data together with their metalinguistic patterns: if we intersect it with the set M of all metalinguistic patterns (by increasing Y until its including set M), or with a subset included in it (Y … M), then we will obtain the total or partial amount of metalinguistic descriptions M' relative to the initial set L' which constitutes its topology :  

L'                 «              Y                 =       M'
[b]:bilabial,         bilabial,phrasal,         bilabial,
[d]:dental,            dental,animate,           dental,
[g]:velar,...           velar,past,......             velar,.....

Before continuing our explanation of topological methods in linguistics, it would be a good idea to answer an implicit question the reader may have: how can linguistics benefit from the fact that the formal object that any linguistic description has to develop is a topological space? Or, putting it in other words: when proving that syntagmatic descriptions of linguistic corpuses are quotient topologies, and that paradigmatic ones are induced topologies, 1) does this verification stop our formal discovery processes or 2) can we go a step further and extract some other relevant conclusions?
If the first possibility holds, then topological linguistics would appear to be an inductive method -at least in their procedures- but always limited to the description of previously known material. If, on the other hand,  the second possibility happens to be true as well, then topological linguistics would behave as a hypothetical-deductive method permitting us to predict  new data and patterns never collected before.
Let me give an example of what I am trying to say. The scientific revolution of the sixteenth century that led to modern physics was obviously thanks to the use of mathematical methods in scientific research. It would be an oversimplification, however, to assume that mathematics had never played a role in natural sciences: in fact medieval physicists, at least since Bacon's time and even before, used to measure empirical data carefully. Nevertheless, it was still an inductive method: reality was tested by the researcher and the results of his investigation were put in a numerical form, but no predictions could be drawn from those large amounts of data. What Galileo Galilei and Isaac Newton really did was to use algebraic equations as a procedure to express the results of previous investigations in a single and elegant manner but, at the same time and most importantly, they made use of those equations as a method for predicting new data and/or patterns  that had never been collected before and that perhaps might never have been reached by empirical procedures. 
The challenge that topological linguistics now faces is the following: can the formal properties of product, quotient, and induced topologies be used to predict new patterns of the text, of the syntagm or of the paradigm respectively, or to confirm that the patterns linguistics usually recognizes are suitable? If so, linguistics would behave as a hypothetical-deductive science. As is known, this was just one of the endeavors of generative grammar when it tried to predict the structures of the sentences of a human language by means of a Boolean algebra system. As far as this aspect is concerned, topological linguistics makes use of hypothetical-deductive methods too, and in that sense it could be considered a branch of generative practice: the difference between both strategies lies in the fact that in the case of topological linguistics prediction is directed neither to metalanguage, nor to language, but to the relationship they maintain. It stands to reason that the results of this approach are unlike those obtained by generative grammar. In the first case the problem does not lie in how such and such an utterance can structurally be described, but rather in what the categories of such a description are, which is sometimes taken for granted by current models. Although I wonder whether the categorization problem can be neglected by any science.
Topological linguistics is not primarily concerned with the description of specific signs (sentences, phrases, words, etc.), but with the setting up of categories that will allow such a description. The fact that generative grammar seems to overlook the establishment of its own foundations has often been criticized. For example, the use of the symbols S(entence), N(oun), Ph(rase), Det(erminer), etc., in grammatical rules without having previously defined them. However, such current criticism seems rather unfair to me: in fact generative grammar begins after categories have been established and its aim is to combine them in categorial chains in order to describe real utterances. What the symbols of a grammar's vocabulary look like constitutes the aim of structuralist descriptions whose interest is to describe the entire linguistic system, rather than its concrete utterances. Topological linguistics benefits from both as it is mainly interested in categorization -like structuralism- and formally predicts -from a typically generative point of view- these categories.
We can use our three fundamental topologies -quotient, product, and induced topologies- in the following way: as each topology conveys a set of formal statements we will try to let the formal laws predict the categories of the linguistic domain that correspond to them. In other words: in spite of taking syntactic categories as previously given entities, and instead of trying to reach them in an inductive manner, we will correlate formal variables with categorial constants and attempt to predict the latter by starting with the former.
As will be seen in the remaining sections:
-A class of formal statements that is usually associated with quotient topologies are the laws of projective geometry: we will use these laws in order to predict the categories of syntax.
-A class of formal statements that is usually associated with induced topologies -derived from the previous quotient topologies- are coordinating procedures: consequently we will use coordinating strategies that usually apply to homogeneous subspaces -such as those that induced topologies are able to yield- to predict how the basic components of phonematics, morphematics, and semematics are established. These components constitute the source of minimal components where the remaining categories of quotient topologies are taken out.    
Nevertheless, the problem of units  or categories  is closely related to the problem of levels   or dimensions: the units of a grammar cannot be put together freely. For example, we cannot add a word to a phoneme and to a clause as in *rabbit + [k] + John came. In fact we can only join elements that belong to the same level or domain of description. Consequently, before we are able to predict categories from formal rules, we need to establish the levels where such categories will apply: namely the dimension problem.