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"[[Topology]]" ([[Fr]]. ''[[topologie]]'') is a branch of [[mathematics]] which deals with the properties of figures in space where are preserved under all continuous deformations.
These properties are those of continuity, 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==
In what have been called his two "topographies" (the first dating from 1900 and the second from 1923), [[Freud]] resorted to [[schema]]s to represent the various parts of the [[psychic apparatus]] and their interrelations.
These schemas implicitly posited an equivalence between [[psychic space]] and [[Euclidean space]].
[[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.<ref>Freud, 1900a: SE V, 536</ref>
His 'first topography' divided the [[psyche]] into three systems: the [[conscious]] (Cs), the [[preconscious]] (Pcs) and the [[unconscious]] (Ucs).
The 'second topography' 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.<ref>{{E}} p.214</ref>
[[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."<ref>{{E}} p.333</ref>
Unlike intuitive images, in which "perception eclipses structure", in [[Lacan]]'s [[topology]] "there is no occultation of the symbolic."<ref>{{E}} p.333</ref>
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[[Lacan]] argues that [[topology]] is not simply a [[metaphor]]ical way of expressing the concept of [[structure]]; it is [[structure]] itself.<ref>Lacan, 1973b</ref>
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 example of a continuous transformation, [[Lacan]] refers to the [[moebius strip]]; just as one passes from one side to the other by following the strip round continuously, 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 transformed into a single loop with very different topological properties; it now has two sides instead of one.
Just as the cut operates a discontinuous transformation in the [[moebius strip]], so an effective [[interpretation]] proferred by the [[analyst]] modifies the [[structure]] of the [[analysand]]'s [[discourse]] in a radical way.
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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]].<ref>Lacan. 1961-2</ref>
Later on, in the 1970s, [[Lacan]] turns his attention to the more complex area of knot theory, especially the [[borromean knot]].
