# Topology

##### Definition

"Topology" 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.

##### Toplogical Space

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.

##### Sigmund Freud

/* In what have been called his two "topographies" (the first dating from 1900 and the second from 1923), Freud resorted to schemas 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.[1] 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 criticizes 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.[2] 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."[3] Unlike intuitive images, in which "perception eclipses structure", in Lacan's topology "there is no occultation of the symbolic."[4]

##### Structure

Lacan argues that topology is not simply a metaphorical way of expressing the concept of structure; it is structure itself.[5] 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.

##### Figures

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.[6] Later on, in the 1970s, Lacan turns his attention to the more complex area of knot theory, especially the Borromean knot.