9) FORMAL LAWS OF INDUCED TOPOLOGIES: THE PARADIGMATIC  COMPONENT

Although quotient topologies have proven to be very useful tools in topological linguistics, there are two more kinds of topology that will also be worthwhile: induced topology and product topology. As said above induced topology is intended to provide systems of categorization for building up the phonemes and the sememes of a given language, i.e. it has to predict how and why some phonetic features are joined in an English phoneme, and how and why the combination of some semantic features yields a sememe in English (or in any other language, for that matter). On the other hand, product topologies will predict the ways the linguistic system is able to merge with other human semiotic or perceptual systems such as social and psychological ones. Briefly, while induced topologies relate to paradigmatic methods of phonematics and semematics, product topologies are concerned with external linguistics. Up to now the induced topology approach has been developed a short step, but the possibilities of product topology remain still undiscovered. 
When we approach the categorization process in  paradigmatic components, it should be apparent that the problem we have to solve is to select some features of reality in spite of some others that sometimes can be very hard to be told apart. For instance, the sounds of a given English utterance can be characterized phonetically by several features, but in fact students only use a few when transcribing it phonemically. Why do we precisely choose for a given phoneme the features we do as its coordinates in English, and do not select any other? If this question would have been raised in any other grammatical domain the answer would have probably been that we select just the features that are shown to work in this domain. But what does really the word to work  mean? A feature is said to work when it is able to control some linguistic processes. For example, the feature [before the verb] contributes to control the functional category of subject in English, and thus if a noun is put behind the verb in standard declarative sentences it cannot be the subject -compare John loves Mary  with Mary loves John, etc. On the contrary, the feature [animate] is less effective in controlling the subject, and hence we can find an inanimate and an animate noun in a sentence without the latter having to become the subject -the Wall Street crack harmed him.
When considering paradigmatic features, either phonic or semic  students usually describe their working capacity in essentially two ways: 
Structuralists like to characterize as important those features that remain even when the conditions of the linguistic pattern get worse. For example, [+animate] is a significant semic feature of girl  in relation to the pattern #______caught a cold# because it explains the impossibility of *the stone caught a cold, but [+human] is not because either the girl  or the dog  are allowed in this slot.
Generativists prefer to use those features that control specific rules. For example, [+animate] is said to be a "selectional restriction" that some verbs share with their subjects in relation to the rule SÆNP+VP.
Nonetheless it is easy to understand that both approaches constitute similar alternative methods of describing linguistic facts rather than two different epistemological statements. In a structuralist description we are interested in the hyperonimous features that remain in positions other than the most contrastive ones, that is, we make use of archiphonemes and archisememes (the archisememe [+animate} is what the girl  and the dog  have in common, for example; this feature, however, does not allow them to alternate in a more contrastive pattern such as the girl  /* the dog took a course in Chemistry, where it is the sememe [+human] that works. In a generativist description these are the very features that yield general selective restriction patterns which govern ruled behavior.
All in all, it seems interesting to develop a method of coordination for phonematics and semematics where only those (phonic or semic) features of reality that behave as Archi(phonemes / sememes) in the given language are taken into account. We will construct our English phonematics, for instance, by making use of the [+labiodental] feature because in English the singular/plural relationship sometimes differs in tenseness degree, but this does not affect the [+labiodental] feature as it is the case in knife/knives, etc (i.e. [+labiodental] can be manifested either as [+labiodental, +tense], or as [+labiodental, +lax]).
Notice that in a Cartesian bidimensional space, and in general in any graphic representation where several axes are taken into account, each unit of the space is defined by a number of values -each relating to one axis- as axes have previously been introduced in the space, no more combinations being available. Thus, in a system such as:

             OY
               Í
              y2------(x1, y2)------(x2, y2)----(x3, y2)
               Í               Í                    Í                    Í
              y1---- -(x1, y1)----   (x2, y1)-----(x3, y1)
               Í               Í                    Í                    Í
             0------ x1 --------- -- x2 ------- --- x3--OX
where there are only two values in OY (y1, y2) and three values in OX (x1, x2, x3), no more than six compound values can be defined, that is (x1, y1), (x2, y1), (x3, y1), (x1, y2), (x2, y2), and (x3, y2).
The above statement suggests a general procedure for paradigmatic coordination that consists of previously looking into the archiunits (archiphonemes and archisememes) that prove to work in the paradigmatic component of a given language, so that we are able to combine them later without repeating any feature of the same axis.
It is no wonder that the axiom which establishes how coordinates are introduced into a topological space -i.e. into an induced topology- that depends on previous definitions  accommodates quite well to the above outlined procedure: 
Let us consider the biunivocal correspondence g´ established between the set C´(A) of possible nodes determined by A and any other pole, and the set C´(B) of possible nodes determined by B and any other pole, i.e. the units where nodes coming from A and nodes coming from B meet:

Let us now establish a biunivocal correspondence g between the elements of an arbitrary abstract set G and the nodes-unit determined by A, i´(A), and another biunivocal correspondence g* of G with the nodes-unit determined by B, i´(B):

set G:{labial, dental, palatal, velar, nasal...} 

The coordinates of any unit of the space would be defined as follows: (a, b) for a unit P where a = g(AP), and b = g*(BP); both coordinates equal -(a, a), (b, b)- for the units where nodes of the same name meet; coordinates (a, ∆) for the units of OB, and (∆, b) for the units of OA. If the space were a n-dimensional one, we would start from n units (A, B, C,...N) and each unit of the space would have received n defining values.
Classical phonematics and componential analysis in semematics perform precisely in this way, although its advocates are usually not aware of the formal background they are employing. If in a given language, as for instance Spanish, a labial archiphoneme B is defined as [+labial, +labial] in syllabic implosive positions where any labial phoneme can replace the other -[ápto] ~ [ábto] ~ [áfto]-, and if a nasal archiphoneme N is also defined as [+nasal, +nasal] in final word position where nasal phonemes are neutralized too -[bém] ~ [bén] ~ [béñ]-, then the phonematist will be able to use two features, the above mentioned having proven to work in Spanish, in order to define the phoneme /m/ of [más] as [+labial, +nasal] for example.

Of course the axiom whose consequences we are trying to discover seems too complicated to bear in mind any time we are describing a phonematic or a semematic paradigm. But the reason for introducing it here serves only to justify the usual procedures  (that consist of putting features together in order to yield units) by showing that if we make sure they are functionally adequate when behaving in neutralization contexts, then they will be supported by the formal laws of induced topology at the same time. However, this kind of support, as odd as it could seem at first glance, is very valuable: as a matter of fact psychological research has shown that humans store phonemic and sememic information in their memory in a coordinated spatial manner (R. League, 1977, Psycholinguistic Matrices, The Hague, Mouton). Once again the formal implication of the procedure outlined in this essay and the Gestalt implication of it prove to be quite related.  

The paradigmatic arrangement of the English vocalic system constitutes a good example of these coordination procedures defined by induced topology. As is known there are a lot of vowels in English which, as a consequence of their reduced space of realization, sometimes show very close pronunciation patterns: there are many words whose pronunciation does not follow a single vocalic pattern, but really joins a diaphonic scale from which only one phonic feature, that characterizes the vowel it is supposed to represent, is retained, whereas the other features of the sound are largely, though not absolutely, free: for example bad  is realized either [bæd] or [bæ:d], direct  vacillates between [di'rekt] and [dai'rekt], room  alternates [rum] and [ru:m] according to individual preferences, etc (D. Jones, 1967, The Phoneme. Its nature and use, Cambridge University Press, Ch. XXVII).
This suggests that English vocalic sounds can appear in three different shapes:
-As diaphones  only if one phonic feature holds permanently, the others belonging to the neighbouring sounds in the articulation space. For example, the diaphone for the vowel of home  would be of the form [o, ∆], where [..∆..] stands for some vocalic shades other than [..o..], and thus [hom], [houm], [ho:um], [höüm], etc; notice that it is not a vowel -i.e. [o]-, but a vocalic shade -i.e. [..o..]-, which constantly remains
-As single phones if the phonic feature that characterizes the vowel is repeated and hence excludes any mixture of other vocalic features. For example the single vowel of let  would be of the form [e, e].
-As double vowel -i.e. as diphthong- if two phonic features combine with each other, neither of them allowing other shades in its space. For example, the double vowel of my  would be of the form [ai] (notice that we consider home, pronounced [houm], as a diaphonic vowel, but my, pronounced [mai], as a diphthong: this is due to the fact that neither [a], nor [i] of [mai] change -i.e. *[mei], *[mau], etc for my -, while the vocalic shade other than [o] in home  certainly does -i.e.[houm], [ho∆m], etc-).
According to A.H. Sommerstein (1977, Modern Phonology, Edward Arnold, 2.1.3) the following vocalic system in English will be composed of either [a, a], [b, b], [a, ∆}, [b, ∆]...etc phonemes -single and diaphonic phones respectively-, or [a,b] phonemes -double ones-: [i:] as in peat , [i] as in pit , [e] as in pet , [æ] as in pat , [a:] as in part , [O] as in pot , [o:] as in port , [U] as in put , [u:] as in boot , [a] as in putt , [œ:] as in pert , [œ] as in the first syllable of parade ; [ei] as in pate , [œu] as in boat , [ai] as in bite , [æu] as in pout , [Oi] as in Boyd , [iœ] as in peered , [eœ] as in pared , [Uœ] as in moored .  
This system is not very common compared to the vocalic systems of other languages. The reason for this lies in the fact that in English vocalic phonematics the archiunits are the single vowels -i. e. [a, a], etc-, and the units the diphthongs -i.e. [a, b], etc-, just the opposite from what happens elsewhere. We can thus conclude that the typologically odd nature of the English vocalic system derives from the oddity of its coordination process, which demonstrates that a strict correlation between the formal and the empirical aspects of these induced topological methods can be established.