Topology

TOPOLOGY (384) [CD]

topology (topologie) Topology (originally called analysis situs by

Leibniz) is a branch of mathematics which deals with the properties of figures

   in space which are preserved under all continuous deformations. These proper-

   ties are those ofcontinuity, contiguity and delimitation. The notion of space in

topology is one of topological space, which is not limited to Euclidean (two-

   and three-dimensional space), nor even to spaces which can be said to have a

   dimension at all. Topological space thus dispenses with all references to

distance, size, area and angle, and is based only on a concept of closeness or

neighbourhood.

     Freud used spatial metaphors to describe the psyche in The Interpretation of

Dreams, where he cites G. T. Fechner's idea that the scene of action of dreams

is different from that of waking ideational life and proposes the concept of

'psychical locality'. Freud is careful to explain that this concept is a purely

topographical one, and must not be confused with physical locality in any

anatomical fashion (Freud, 1900a: SE V, 536). His 'first topography' (usually

referred to in English as 'the topographic system') divided the psyche into

three systems: the conscious (Cs), the preconscious (Pcs) and the unconscious

(Ucs). The 'second topography' (usually referred to in English as 'the struc-

tural system') divided the psyche into the three agencies of the ego, the

superego and the id.

     Lacan criticises these models for not being topological enough. He argues

that the diagram with which Freud had illustrated his second topology in The

Ego and the Id (1923b) led the majority of Freud's readers to forget the

analysis on which it was based because of the intuitive power of the image

(see E, 214). Lacan's interest in topology arises, then, because he sees it as

providing a non-intuitive, purely intellectual means of expressing the concept

of STRUCTURE that is so important to his focus on the symbolic order. It is thus

the task of Lacan's topological models 'to forbid imaginary capture' (E, 333).

Unlike intuitive images, in which 'perception eclipses structure', in Lacan's

topology 'there is no occultation of the symbolic' (E, 333).

    Lacan argues that topology is not simply a metaphorical way of expressing

the concept of structure; it is structure itself (Lacan, 1973b). He emphasises

that topology privileges the function of the cut (coupure), since the cut is what

distinguishes a discontinuous transformation from a continuous one. Both

kinds of transformation play a role in psychoanalytic treatment. As an exam-

ple of a continuous transformation, Lacan refers to the MOEBIUS STRIP; j¸St SS

one passes from one side to the other by following the strip round continu-

ously, so the subject can traverse the fantasy without making a mythical leap

from inside to outside. As an example of a discontinous transformation, Lacan

also refers to the moebius strip, which when cut down the middle is trans-

formed into a single loop with very different topological properties; it now has

two sides instead of one. Just as the cut operates a discontinuous transforma-

tion in the moebius strip, so an effective interpretation proferred by the analyst

modifies the structure of the analysand's discourse in a radical way.

    While SCHEMA L and the other schemata which are produced in the 1950s can

be seen as Lacan's first incursion into topology, topological forms only come

into prominence when, in the 1960s, he turns his attention to the figures of the

TORUs, the moebius strip, Klein's bottle, and the cross-cap (see Lacan, 1961-2).

Later on, in the 1970s, Lacan turns his attention to the more complex area of

knot theory, especially the BORROMEAN KNOT. For an introduction to Lacan's use

of topological figures, see Granon-Lafont (1985).

topology, 22, 34, 74, 89-90, 131, 144, 147, 155-6, 161, 164, 181-2, 184, 203, 206, 209, * 235, 244-5, 257, 270-1 Seminar XI