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Topology

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{{Top}}[[topologie]]{{Bottom}}
=====Definition=====
"[[Topology]]" is a branch of [[mathematics]] which deals with the properties of [[figures]] in [[topology|space]] where are preserved under all continuous deformations. These properties are those of continuity, contiguity and delimitation.
=====Toplogical Space=====The [[Topologynotion]] refers primarily to the branch of [[mathematicstopology|space]] in [[topology]] that rigorously treats questions is one of neighborhoods[[topology|topological space]], limitswhich is not limited to Euclidean (two- and [[three]]-dimensional [[space]]), nor even to spaces which can be said to have a [[dimension]] at all. [[topology|Topological space]] thus dispenses with all references to distance, size, area and continuityangle, and is based only on a [[concept]] of closeness or neighbourhood.
=====Sigmund Freud=====/* In what have been called his two "[[Psychoanalyststopology|topographies]] have applied it " (the first dating from 1900 and the second from 1923), [[Freud]] resorted to [[schema]]s to [[represent]] the study various parts of the [[unconsciouspsychic apparatus]] and their interrelations. These schemas implicitly posited an equivalence between [[structurepsychic]]sspace 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 "[[topology|first topography]]" [[divided]] the [[psyche]] into three systems: the [[conscious]] (Cs), the [[preconscious]] ([[Pcs]]) and the [[unconscious]] ([[Ucs]]). The "[[topology|second topography]]" divided the [[psyche]] into the three [[agencies]] of the [[ego]], the [[superego]] and the [[id]].
[[TopologyLacan]] (criticizes these models for not [[being]] [[topological]] enough. He argues that the diagram with which [[Freud]] had illustrated his second topology in ''topologie[[The Ego and the Id]]''(1923b) is a branch led the majority of [[mathematicsFreud]]'s readers to forget the [[analysis]] on which deals with it was based because of the properties intuitive [[power]] of figures in space which are preserved under all continuous deformationsthe [[image]].These properties are those of continuity, contiguity and delimitation<ref>{{E}} p. The notion of space 214</ref> [[Lacan]]'s interest in [[topology is one of topological space]] arises, then, which is not limited to Euclidean (twobecause he sees it as providing a non- and three-dimensional space)intuitive, nor even purely [[intellectual]] means of expressing the concept of [[structure]] that is so important to spaces which can be said his focus on the [[symbolic order]]. It is thus the task of [[Lacan]]'s topological models "to have a dimension at allforbid [[imaginary]] [[capture]]."<ref>{{E}} p. 333</ref> Topological space thus dispenses with all references to distanceUnlike intuitive [[images]], sizein which "[[perception]] eclipses structure", area and angle, and in [[Lacan]]'s [[topology]] "there is based only on a concept no occultation of closeness or neighbourhoodthe [[symbolic]]."<ref>{{E}} p.333</ref>
=====Structure=====
[[Lacan]] argues that [[topology]] is not simply a [[metaphor]]ical way of expressing the concept of [[structure]]; it is [[structure]] itself.<ref>{{L}} "[[Works of Jacques Lacan|L'Étourdit]]," ''[[Scilicet]]'', no. 4, 1973: pp. 5-52</ref> He emphasizes 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.
==Freud===Figures=====In what have been called his two "topographies" (the first dating from 1900 While [[schema L]] and the second from 1923), other [[Freudschemata]] resorted to which are produced in the 1950s can be seen as [[schemaLacan]]'s first incursion into [[topology]], topological forms only come into prominence when, in the 1960s, he turns his attention to represent the various parts figures of the [[psychic apparatustorus]], the [[moebius strip]], [[Klein]] 's bottle, and their interrelationsthe [[cross-cap]].<ref>{{L}} ''[[Works of Jacques Lacan|Le Séminaire. Livre IX. L'identification, 1961-62]]'', unpublished. These schemas implicitly posited an equivalence between </ref> Later on, in the 1970s, [[Lacan]] turns his attention to the more [[complex]] area of [[knot]] [[psychic spacetheory]] and , especially the [[Euclidean spaceBorromean knot]].
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==Early on, [[Jacques Lacan]] noted that the limitations of such a naive topology had restricted [[Freudian theory]], not only in the description of the [[psychic apparatus]] (a description that in the end required an appeal to the economic point of view), but also in the specificity of [[clinical structure]]s.The hypothesis that [[the unconscious is structured like a language,]] that is, in two dimensions, led [[Lacan]] to the [[topology]] of [[surface]]s. The concept of '[[foreclosure]]', for example, which he constructed on the basis of this [[topology]], confirmed the heuristic value of his approach. In his 1961-1962 seminar "[[L%27identification|Identification]]", [[Lacan]] unveiled a collection of [[topology|topological]] [[object]]s — such as the [[torus]], the [[Möbius strip]], and the [[cross-cap]] — that served pedagogical aims. But already he saw them as more than just models. With the ]]Borromean knot\\, introduced in 1973, he took the position that these [[object]]s were a real presentation of the [[subject]] and not just a [[representation]].  Below are several of [[Lacan]]'s [[topology|topological]] [[object]]s. ==The Cut and the Signifier== Far from being given a priori, every space is organized on the basis of cuts and can actually be considered as a cut in the space of a higher dimension. We are familiar with the subjective impact of this: The events of our lives only become history through the castration complex, which organizes our reality at the price of an imaginary cutting off of the penis. According to Freud, by introjecting a single trait of another, the subject identifies with the other (at the price of losing this person as a love object). In the single trait Lacan found the very structure of the signifier: A cut allows the lost object to fall away. He called this cut the "unary trait." The linguist Ferdinand de Saussure insisted on the fundamentally negative, purely differential character of the signifier. Lacan formalized this property in the double loop, or "interior eight," in which the gap created by the cut is closed after a second trip around a fictional axis. The difference of the signifier from itself is indicated by the difference between the two trips around the loop (Figure 1). ==The Möbius Strip and Interpretation== If a signifier represents the subject for another signifier, then the subject would be supported by a surface whose edge would be a signifying cut. Note that the plane—the usual screen for the subject's images, figures, and dreams, that is, plans—is a surface that does not meet these conditions. The double loop cannot be drawn on a plane without showing a cut. The same is true of a sphere, a simple representation of the universe. The Möbius strip, on the other hand, can represent this cut and symbolize the subject of the unconscious. Since a Möbius strip only has one surface, it is possible to pass from one side to the other without crossing over any edge—an apt representation of the return of the repressed. The Möbius strip also has certain other peculiarities. A cut that runs one-third from the edge and parallel to the edge divides the strip into a two-sided strip linked to what remains of the original Möbius strip. But if this cut is made in the center, it does not divide the Möbius strip in two. Instead, the entire strip is transformed into a strip with two sides. This characteristic illustrates the equivalence between the Möbius strip (the subject) and the medial cut that transforms it, and also provides a model of how interpretation functions. Interpretation does not abolish the unconscious. On the contrary, it makes the unconscious real for the subject by its transformed appearance as another (an Other) surface (figure 2).  ==The Torus=See Also= Lacan made different uses of the torus. By drawing Venn diagrams, traditionally used to illustrate basic logical operations, on the surface of the torus, he demonstrated the extent to which our thinking depends upon the plane surface, and he also provided another possible basis for the logic of the unconscious (Figure 3). By inscribing the same circles on the surface of the torus, Lacan revealed the logic of the unconscious discovered by Freud (Figure 4). On the torus, only symmetrical difference is consistent. Thus we have a demonstration of how the signifier can be different from all other signifiers and also from itself. Lacan also used the torus to represent the subject as the subject of demand. In this sense, the torus can be conceived as the surface created by the iteration of the trajectory of the subject's demand. This trajectory turns around two different empty spaces, one that is "internal," D, the lack created in the real by speech, and one that is "central," d, corresponding to the place of the elusive object of desire that the drive goes around before completing the loop (Figure 5). For every torus, there is a complementary torus, and the empty spaces of the two are the inverse of each other. Lacan made this structure of complementary toruses the support of the neurotic illusion that makes the demand of the Other the object of subject's desire and, conversely, makes the desire of the Other the object of subject's demand. This structure also arises from the fact that on a torus, the signifying cut (the double loop) does not detach any fragment. Neurotic subjects, insofar as they give in to neurosis, insofar as they are "in the torus," are not organized around their own castration, but instead excuse themselves by substituting the Other's demand for the object of their fantasy (figure 6).  ==The Cross-Cap== The cross-cap, or more precisely, the projective plane, can represent the subject of desire in relation to the lost object. A double loop drawn on its surface in effect divides this single-sided surface into two heterogeneous parts: a Möbius strip representing the subject and a disk representing object a, the cause of desire. The disk is centered on a point that is related to the irreducible singularity of this surface, which Lacan identified with the phallus. Unlike the representation of the subject produced on the torus, here a single cut, which symbolizes castration, produces both the subject and the object in its divisions (figure 7).  ==The Borromean Knot== Introduced by Lacan in 1973, the Borromean knot is the solution to a problem perceivable only in Lacanian theory but having extremely practical clinical applications. The problem is: How are the three registers posited as making up subjectivity—the real (R), the symbolic (S), and the imaginary (I)—held together? Indeed, the symbolic (the signifier) and the imaginary (meaning) seem to have hardly anything in common—a fact demonstrated by the abundance and heterogeneity of languages. Moreover, the real, by definition, escapes the symbolic and the imaginary, since its resistance to them is precisely what makes it real. This is why Lacan identified the real with the impossible.) In psychoanalysis, the real resists, and thus is distinct from, the imaginary defenses that the ego uses specifically to misrecognize the impossible and its consequences. If each of the three registers R, S, and I that make up the Borromean knot is recognized to be toric in structure and the knot is constructed in three-dimensional space, it constitutes the perfect answer to the problem above, because it realizes a three-way joining of all three toruses, while none of them is actually linked to any other: If any one of them is cut, the other two are set free. Reciprocally, any knot that meets these conditions is called Borromean. Note that the subject is now defined by such a knot and not merely, as with the cross-cap, as the effect of a cut (figure 8). Unfortunately, this ideal solution, which could be considered normal (without symptoms), seems to lead to paranoia. Lacan considered this to be the result of failure to distinguish among the three registers, as if they were continuous, which indeed occurs in clinical work. Being identical, R, S, and I are only differentiated by means of a "complication," a fourth ring that Lacan called the "sinthome." By making a ring with the three others, the sinthome (symptom) differentiates the three others by assuring their knotting (figure 9). In this arrangement, the sinthome has the function of determining one of the rings. If it is attached to the symbolic, it plays the role of the paternal metaphor and its corollary, a neurotic symptom. Lacan also drew upon non-Borromean knots, generated by "slips," or mistakes, in tying the knots. These allowed him to represent the status of subjects who are unattached to the imaginary or the real and who compensate for this with supplements (Lacan, 2001). In such cases the sinthome is maintained. By using knots, Lacan was able to reveal his ongoing research without hiding its uncertainties. The value of the knots, which resist imaginary representation, is that they advance research that is not mere speculation and that they can grasp—at the cost of abandoning a grand synthesis—a few "bits of the real" (Lacan, 1976-1977, session of March 16, 1976). Even though he knew something about topology as practiced by mathematicians, Lacan advised his students "to use it stupidly" (Lacan, 1974-1975, session of December 17, 1974) as a remedy for our imaginary simplemindedness. He also recommended manually working with the knots by cutting surfaces and tying knots. Finally, for Lacan, topology had not only heuristic value but also valuable implications for psychoanalytic practice.  ==More== 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>see E, 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, 333</ref> Unlike intuitive images, in which "perception eclipses structure", in Lacan's topology "there is no occultation of the symbolic."<ref>E, 333</ref> Lacan argues that topology is not simply a metaphorical 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. 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>see 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]].<ref>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]]</ref> =={{See Also==}}* [[Knot]]* [[Schema L]]* [[Schema R]] * [[Seminars]] * [[Thalassa. A Theory of Genitality]]* [[Unary trait]]* [[torus]]* [[borromean Borromean knot]]* [[extimacyMoebius strip]]* [[subject]]* [[cross-cap]]* [[knot]]{{Also}}
==References==
<references/>
# Lacan, Jacques. (1975). La troisième, intervention de J. Lacan, le 31 octobre 1974. Lettres de l 'École Freudienne, 16, 178-203.# Lacan, Jacques. (1974-1975). Le séminaire, livre XXII, R.S.I. Ornicar? 2-5.# Lacan, Jacques. (1976-1977). Le séminaire XXIII, 1975-76[[Category: Le sinthome. Ornicar? 6-11.# Lacan, Jacques. (2001). Joyce: Le symptôme. In his Autres écrits. Paris: Seuil.Psychoanalysis]]
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