# Topology

# TOPOLOGY

*Topology* refers primarily to the branch of mathematics that rigorously treats questions of neighborhoods, limits, and continuity. Psychoanalysts have applied it to the study of unconscious structures.

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 and Euclidean space.

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 structures. The hypothesis that the unconscious is structured like a language, that is, in two dimensions, led Lacan to the topology of surfaces. The concept of foreclosure, for example, which he constructed on the basis of this topology, confirmed the heuristic value of his approach.

In his seminar "Identification" (1961-1962), Lacan unveiled a collection of topological objects—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 objects were a real presentation of the subject and not just a representation. Below are several of Lacan's topological objects.

## 1. 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).

## 2. 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).

## 3. The Torus

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).

## 4. 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).

## 5. 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.

Bernard Vandermersch

*See also:* Knot; L and R schemas; Seminar, Lacan's; Signifier/signified; Structural theories; Symptom/sinthome; *Thalassa. ATheory of Genitality* ; Unary trait.

## Bibliography

Bourbaki, Nicolas. (1994). *Elements of the history of mathematics* (John Meldrum, Trans.). Berlin: Springer-Verlag.

Darmon, Marc. (1990). *Essais sur la topologie Lacanienne*. Paris:Éditions de l'Association Freudienne Internationale.

Lacan, Jacques. (1975). La troisième, intervention de J. Lacan, le 31 octobre 1974. *Lettres de l* '*École Freudienne*, *16*, 178-203.

——. (1974-1975). Le séminaire, livre XXII, R.S.I. *Ornicar? 2-5*.

——. (1976-1977). Le séminaire XXIII, 1975-76: Le sinthome. *Ornicar? 6-11*.

——. (2001). Joyce: Le symptôme. In his *Autres écrits*. Paris: Seuil.

Pont, Jean-Claude. (1974). *La topologie algébrique des origines à Poincaré*. Paris: Presses Universitaires de France.

# Topology

# Topology

Topology, which is often described as “rubber-sheet geometry,” is a branch of geometry that focuses on distortion. Topology describes mathematically the

features of a geometric shape that do not change when the shape is twisted, stretched, or squeezed. Tearing, cutting, and combining shapes do not apply to topology. Topology helps to solve problems about determining the number of colors necessary to illustrate maps, about distinguishing the characteristics of knots, and about understanding the structure and behavior of DNA molecules.

## Topological equivalency

The crucial problem in topology is deciding when two shapes are equivalent. Unlike Euclidean geometry, which focuses on the measurement of distances between points on a shape, topology focuses on the similarity and continuity of certain features of geometrical shapes. For example, in Figure 1, each of the two shapes has five points: a through e. The sequence of the points does not change from shape 1 to shape 2, even though the distance between the points, for example, between points b and d, changes significantly because shape 2 has been stretched. Thus the two shapes in Figure 1 are topologically equivalent, even if their measurements are different.

Similarly, in Figure 2, each of the closed shapes is curved, but shape 3 is more circular, and shape 4 is a flattened circle, or ellipse. However, every point on shape 3 can be mapped or transposed onto shape 4.

Shapes 1 and 2 are both topologically equivalent to each other, as are shapes 3 and 4. That is, if each were a rubber band, it could be stretched or twisted into the same shape as the other without connecting or disconnecting any of its points. However, if either of the shapes in each pair is torn or cut, or if any of the points in each pair join together, then the shapes are not topologically equivalent. In Figure 3, neither of the shapes is topologically equivalent to any of the

shapes in Figures 1 or 2, nor are shapes 5 and 6 equivalent to each other. The circles in shape 5 are fused; and the triangle in shape 6 has a broken line hanging from its apex.

## Famous topologists

Topological ideas can be traced back to Gottfried Wilhelm Leibniz (1646-1716), but three of the most famous figures in the development of topology are Möbius, Riemann, and Klein.

Augustus Ferdinand Möbius (1790-1868) is best known for his invention of the Möbius strip, which is a simple strip of paper that is twisted and connected so that it has only one side (Figure 4). Normally, cutting a strip of paper into a long, narrow rectangle and connecting the ends will result in a belt-like loop with two sides. A person cannot draw a single line with a pencil on both sides of the belt-like loop without crossing an edge. In constructing the Möbius strip, however, the strip of paper is twisted as it is looped, and the result is a one-sided object.

At first, this one-sided construction seems impossible, but if a person draws a straight, continuous line on the Möbius strip, the line will cover the entire length of both sides of the strip without ever crossing an edge, and it will return to its starting point in one long stroke.

Georg Friedrich Bernhard Riemann (1826-1866) developed some of the most important topological ideas about the stretching, bending, and twisting of surfaces, but he died prematurely at the age of 39 before he could expand significantly upon his ideas.

Felix Klein (1849-1925) is best known for the paradoxical figure which was named after him: the Klein bottle.

The Klein bottle is a one-sided object that has no edge. It is a tapered tube whose neck is bent around to enter the side of the bottle. The neck continues into the base of the bottle where it flares out and rounds off to form the outer surface of the bottle (Figure 5). Like the Möbius strip, any two points on the bottle can be joined by a continuous line without crossing an edge, which gives the impression that the inside and outside of the Klein bottle are continuous.

## Classifications

topological shapes are classified according to how many holes they have. Shapes with no holes at all-spheres, eggs, and convex or concave shapes like bowls—are regarded as genus (or type) 0 shapes.

Genus 1 shapes have one hole in them: a donut (or torus), a wedding band, a pipe, or anything with a looped handle (a teacup). Genus 2 shapes have two holes in them, for example, a figure eight. Genus 3 shapes (most pretzels) have three holes in them. And so on.

The determining feature in classifying the topological genus of a shape is deciding if every point in one shape can be transposed or mapped onto a point in the other. Sometimes this process is easy, as in the case of a wedding ring and a donut, which are genus 1 topological shapes. But with complex genus shapes of 4 and above, the determination can be difficult.

## Current research

Topology has a number of interesting applications, including molecular biology and synthesizing new chemical compounds to help in gene therapy. For example, strands of DNA (deoxyribonucleic acid, which contains the genetic code that defines life) often become knotted. Researchers need to know if the knotted mass of DNA is just one strand of DNA that has wound back upon itself, or if it is several strands of DNA which have become entangled. Topology, especially knot theory, helps molecular biologists solve such problems of equivalency.

Topology also has applications in synthesizing new molecules, called dendrimers, which may be used in gene therapy. Dendrimers are branching, tree like polymers (strings of smaller molecules) synthesized in laboratories by chemically attaching the polymers to a core molecule. Dendrimers are approximately the same size as many naturally occurring molecules, including DNA and proteins. In gene therapy, new genetic material needs to be transported to the nucleus of a cell to replace damaged or defective genes in human cells. Then the repaired cells can reproduce more healthy cells. However, there must be a way to transport the healthy DNA into cells, which is where dendrimers come in. Normally, DNA wraps itself around clusters of proteins called histones. Gene therapists can fool the immune system of the human body by substituting dendrimers for naturally occurring clusters of histones and thus transport healthy DNA into damaged cells. Topology is useful in this process, because its purpose is to decide mathematically which shapes are equivalent. The closer that synthetic dendrimers mimic the naturally-occurring histones, then the greater are the chances that the DNA will be transported to create healthy cells.

*See also* Polymer.

### KEY TERMS

**Dendrimer molecules** —Branching, treelike polymers synthesized in laboratories by attaching the polymers to a core molecule.

**DNA** —Deoxyribonucleic acid, a thin, ladderlike molecular structure which contains the genetic code that defines life.

**Euclidean geometry** —Geometry based upon the postulates of the Greek educator, Euclid who lived roughly 300 BC Euclid’s parallel postulate that only one line may be drawn through a given point such that it is parallel to another line, was overthrown in the nineteenth century and gave way to the nonEuclidean geometries created by Gauss and Riemann.

**Geometry** —The area of mathematics that deals with deductions concerning the measurements and relationships between points, lines, angles, and figures in physical space.

**Polymers** —Strings of smaller molecules.

## Resources

### BOOKS

Ball, W.W. Rouse. *A Short Account of the History of Mathematics*. London: Sterling Publications, 2002.

Patrick Moore

# Topology

# Topology

In the social sciences, it is important to know if close-by models of human behavior and interaction entail close-by predictions. More precisely, the question is whether a sequence of models, which mirror reality in an increasingly accurate manner, yield predictions that converge to those that can be observed in the real world. *Topology* is a mathematical structure designed to express robustness, approximation, convergence, and continuity, and is thus useful in determining the relevance of a given social-science model.

For example, economies differ according to the endowments of individuals, the available production technologies, and the preferences of individuals over bundles of commodities. General Equilibrium Theory predicts the market prices that can emerge as a result of various combinations of these factors. The robustness of these predictions can then be tested through a consideration of a *topological space* of economies in which one can examine whether market prices change continuously with changes in the economies’ characteristics.

Similarly, in Game Theory, Nash equilibrium is a prediction of the strategies that players will choose noncooperatively as a function of their preferences over the outcomes entailed by strategy profiles. When considering a topological space of games, one can check how Nash equilibria vary with the game specification. Alternatively, Social Choice Theory, based on normative considerations, prescribes a strategy profile that the players *should* choose collectively (rather than noncooperatively) in the social situation at hand. In a topological space of such social choice problems, one can check if close-by behavior is prescribed in close-by situations.

Formally, given a space *X* of model characteristics (or some other objects of interest), a topology is a system *T* of *open sets,* which are subsets of *X* with the following properties: (1) The union of any collection of open sets is open; (2) the intersection of a finite number of open sets is open; and (3) both the entire space and the empty set are open. The complement of an open set is called *close*. A space *X* equipped with a topology of open subsets *T* is called a *topological space*.

A pertinent example is the case in which the space is *metric,* i.e., when there exists a metric that defines the distance between any two objects in the space (such that the distance of an object to itself is zero; the distance from *x* to *y* is the same as the distance from *y* to *x* ; and the distance from *x* to *z* is no larger than the distance from *x* to *y* plus the distance from *y* to *z* ). In such a case, the unions of “open balls” constitute a topology (an open ball of radius *r* around a point *x* in the space is the set of points in the space whose distance to *x* is smaller than *r* ). Moreover, a set *v* is open if and only if every point *x* ∊ *V* has an open ball around it contained in *V*.

Hence, if for some property *P* of the model, the set *V* of characteristics at which *P* obtains is open, then the property *P* is *robust :* When a characteristic *x* ∊ *V* is measured with a small enough error, the measurement will still have the property *P*.

The idea of robustness, as captured by open sets, carries over also to families of economic models whose topological structure is so rich it cannot always be compatible with a metric. Financial models of dynamic investment in continuous time, and stochastic uncertainty—over objective circumstances, as well as over others’ uncertainties— are two important examples.

A sequence *x* of points in the space *converges* to the point *x* if for every open set *V* containing *x* there exists a stage *N* beyond which all points *x _{n}, n ≥ N* in the sequence belong to

*V*.

This definition of convergence applies not only to sequences, but also to nets. In a *net x ,* the indices *n* are not necessarily the natural numbers. Rather, they may form a *directed system* —a set where not every pair of distinct indices *n, n′* is characterized by one of the indices being larger than the other, but where there always exists another index *n″* that is larger than them both.

Convergence of sequences and nets depends on the richness of the topology. In the *trivial topology,* containing only the empty set and the entire space *X,* every net converges to every point. At the other extreme, with the *discrete topology,* in which every subset is open (and, in particular, every subset containing a single point is open), a net *x _{n}* converges to

*x*only if for some index

*N*and onward

*x*for all

_{n}= x*n ≥ N*. Hence, the choice of topology expresses the extent to which the modeler views different points (or objects or model characteristics) in the space as distinct or similar. The more fine-detailed the distinctions are, the richer will be the topology, and fewer nets will be converging to any given point

*x*.

When *X* and *Y* are topological spaces, we say that the function *f* : *X* → *Y* is *continuous at the point x* ∊ *x* if for every net *x _{n}* converging to

*x,*the net

*f(x*) converges to the point

_{n}*f (x)*∊

*Y*. We say that

*f*is

*continuous if it*is continuous at every point

*x*∊

*X*

If *X* is a space of model characteristics and *Y* is a space of potential predictions, one would like the prediction function *f* : *X* → *Y* of the model to be continuous— otherwise a slight misspecification of the characteristics might yield wildly distinct predictions.

One can show that *f* is continuous if and only if for every open set *W* ⊆ *Y,* the set *f* ^{–1} ( *W) = {x* ∊ *X f(x)* ∊ *W}* is open in *X* Thus, the richer is the topology of *X* and the poorer is the topology of *Y,* the more functions *f* from *X* to *Y* are continuous.

**SEE ALSO** *Manifolds*

## BIBLIOGRAPHY

Aliprantis, Charalambos D., and Kim C. Border. 2006. *Infinite Dimensional Analysis: A Hitchhiker’s Guide*. 3rd ed. Berlin and New York: Springer.

Royden, Halsey. 1988. *Real Analysis*. 3rd ed. Englewood Cliffs, NJ: Prentice-Hall.

*Aviad Heifetz*

# Topology

# Topology

Topology, which is often described as "rubber-sheet geometry," is a branch of **geometry** that focuses on distortion. Topology describes mathematically the features of a geometric shape that do not change when the shape is twisted, stretched, or squeezed. Tearing, cutting, and combining shapes do not apply to topology. Topology helps to solve problems about determining the number of colors necessary to illustrate maps, about distinguishing the characteristics of knots, and about understanding the structure and behavior of DNA molecules.

## Topological equivalency

The crucial problem in topology is deciding when two shapes are equivalent. Unlike Euclidean geometry, which focuses on the measurement of distances between points on a shape, topology focuses on the similarity and **continuity** of certain features of geometrical shapes. For example, in Figure 1, each of the two shapes has five points: a through e. The sequence of the points does not change from shape 1 to shape 2, even though the distance between the points, for example, between points b and d, changes significantly because shape 2 has been stretched. Thus the two shapes in Figure 1 are topologically equivalent, even if their measurements are different.

Similarly, in Figure 2, each of the closed shapes is curved, but shape 3 is more circular, and shape 4 is a
flattened **circle** , or **ellipse** . However, every point on shape 3 can be mapped or transposed onto shape 4.

Shapes 1 and 2 are both topologically equivalent to each other, as are shapes 3 and 4. That is, if each were a rubber band, it could be stretched or twisted into the same shape as the other without connecting or disconnecting any of its points. However, if either of the shapes in each pair is torn or cut, or if any of the points in each pair join together, then the shapes are not topologically equivalent. In Figure 3, neither of the shapes is topologically equivalent to any of the shapes in Figures 1 or 2, nor are shapes 5 and 6 equivalent to each other. The circles in shape 5 are fused; and the triangle in shape 6 has a broken line hanging from its apex.

## Famous topologists

Topological ideas can be traced back to Gottfried Wilhelm Leibniz (1646-1716), but three of the most famous figures in the development of topology are Möbius, Riemann, and Klein.

Augustus Ferdinand Möbius (1790-1868) is best known for his invention of the **Möbius strip** , which is a simple strip of **paper** that is twisted and connected so that it has only one side. Normally, cutting a strip of paper into a long, narrow **rectangle** and connecting the ends will result in a belt-like loop with two sides. A person cannot draw a single line with a pencil on both sides of the belt-like loop without crossing an edge. In constructing the Möbius strip, however, the strip of paper is twisted as it is looped, and the result is a one-sided object.

At first, this one-sided construction seems impossible, but if a person draws a straight, continuous line on the Möbius strip, the line will cover the entire length of both sides of the strip without ever crossing an edge, and it will return to its starting point in one long stroke.

Georg Friedrich Bernhard Riemann (1826-1866) developed some of the most important topological ideas about the stretching, bending, and twisting of surfaces, but he died prematurely at the age of 39 before he could expand significantly upon his ideas.

Felix Klein (1849-1925) is best known for the paradoxical figure which was named after him: the Klein bottle.

The Klein bottle is a one-sided object that has no edge. It is a tapered tube whose neck is bent around to enter the side of the bottle. The neck continues into the base of the bottle where it flares out and rounds off to form the outer surface of the bottle. Like the Möbius strip, any two points on the bottle can be joined by a continuous line without crossing an edge, which gives the impression that the inside and outside of the Klein bottle are continuous.

## Classifications

Topological shapes are classified according to how many holes they have. Shapes with no holes at all-spheres, eggs, and convex or concave shapes like bowls—are regarded as genus (or type) 0 shapes.

Genus 1 shapes have one hole in them: a donut (or **torus** ), a wedding band, a pipe, or anything with a looped handle (a teacup). Genus 2 shapes have two holes in them, for example, a figure eight. Genus 3 shapes (most pretzels) have three holes in them. And so on.

The determining feature in classifying the topological genus of a shape is deciding if every point in one shape can be transposed or mapped onto a point in the other. Sometimes this process is easy, as in the case of a wedding ring and a donut, which are genus 1 topological shapes. But with complex genus shapes of 4 and above, the determination can be difficult.

## Current research

Topology has a number of interesting applications, including **molecular biology** and synthesizing new chemical compounds to help in **gene therapy** . For example, strands of DNA (deoxyribonucleic acid, which contains the genetic code that defines life) often become knotted. Researchers need to know if the knotted mass of DNA is just one strand of DNA that has wound back upon itself, or if it is several strands of DNA which have become entangled. Topology, especially knot theory, helps molecular biologists solve such problems of equivalency.

Topology also has applications in synthesizing new molecules, called dendrimers, which may be used in **gene** therapy. Dendrimers are branching, tree-like polymers (strings of smaller molecules) synthesized in laboratories by chemically attaching the polymers to a core **molecule** . Dendrimers are approximately the same size as many naturally occurring molecules, including DNA and **proteins** . In gene therapy, new genetic material needs to be transported to the nucleus of a **cell** to replace damaged or defective genes in human cells. Then the repaired cells can reproduce more healthy cells. However, there must be a way to transport the healthy DNA into cells, which is where dendrimers come in. Normally, DNA wraps itself around clusters of proteins called histones. Gene therapists can fool the **immune system** of the human body by substituting dendrimers for naturally occurring clusters of histones and thus transport healthy DNA into damaged cells. Topology is useful in this process, because its purpose is to decide mathematically which shapes are equivalent. The closer that synthetic dendrimers mimic the naturally-occurring histones, then the greater are the chances that the DNA will be transported to create healthy cells.

See also Polymer.

## Resources

### books

Ball, W.W. Rouse. *A Short Account of the History of Mathematics.* London: Sterling Publications, 2002.

Kline, Morris. *Mathematical Thought: From Ancient to Modern Times.* 3 vols. New York: Oxford University Press, 1972.

Peterson, Ivars. *The Mathematical Tourist: Snapshots of Modern Mathematics.* New York: W. H. Freeman,1988.

### periodicals

Tomalia, Donald A. "Dendrimer Molecules." *Scientific American* 272 (May 1995): 62-66.

Patrick Moore

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Dendrimer molecules**—Branching, treelike polymers synthesized in laboratories by attaching the polymers to a core molecule.

**DNA**—Deoxyribonucleic acid, a thin, ladderlike molecular structure which contains the genetic code that defines life.

**Euclidean geometry**—Geometry based upon the postulates of the Greek educator, Euclid who lived roughly 300 b.c. Euclid's parallel postulate that only one line may be drawn through a given point such that it is parallel to another line, was overthrown in the nineteenth century and gave way to the nonEuclidean geometries created by Gauss and Riemann.

**Geometry**—The area of mathematics that deals with deductions concerning the measurements and relationships between points, lines, angles, and figures in physical space.

**Polymers**—Strings of smaller molecules.

# Topology

# Topology

Topology is a branch of mathematics sometimes known as rubber-sheet geometry. It deals with the properties of a geometric figure that do not change when the shape is twisted, stretched, or squeezed. In topological studies, the tearing, cutting, and combining of shapes is not allowed. The geometric figure must stay intact while being studied. Topology has been used to solve problems concerning the number of colors necessary to illustrate maps, about distinguishing the characteristics of knots, and about understanding the structure and behavior of DNA (deoxyribonucleic acid) molecules, which are responsible for the transferring of physical characteristics from parents to offspring.

## Topological equivalency

The crucial problem in topology is deciding when two shapes are equivalent. The term equivalent has a somewhat different meaning in topology than in Euclidean geometry. In Euclidean geometry, one is concerned with the measurement of distances and angles. It is, therefore, a form of quantitative analysis. In contrast, topology is concerned with similarities in shape and continuity between two figures. As a result, it is a form of qualitative analysis.

For example, in Figure 1 on page 1898, each of the two shapes has five points: a through e. The sequence of the points does not change from shape 1 to shape 2, even though the distance between the points changes. Thus the two shapes in Figure 1 are topologically equivalent, even though their measurements are different.

Similarly, in Figure 2, each of the closed shapes is curved, but shape 3 is more circular, and shape 4 is a flattened circle, or ellipse. However, every point on shape 3 can be mapped or transposed onto shape 4. So the two figures are topologically equivalent to each other.

Shapes 1 and 2 are both topologically equivalent to each other, as are shapes 3 and 4. That is, if each were a rubber band, it could be stretched or twisted into the same shape as the other without connecting or disconnecting any of its points. However, if either of the shapes in each pair is torn or cut, or if any of the points in each pair join together, then the shapes are not topologically equivalent. In Figure 3, neither of the shapes is topologically equivalent to any of the shapes in Figures 1 or 2, nor are shapes 5 and 6 equivalent to each other. The circles in shape 5 are fused, and the triangle in shape 6 has a broken line hanging from its apex.

## Famous topologists

Topological ideas can be traced back to German mathematician Gottfried Wilhelm Leibniz (1646–1716). However, three of the most famous figures in the development of topology are later German mathematicians: Augustus Ferdinand Möbius (1790–1868), Georg Friedrich Bernhard Riemann (1826–1866), and Felix Klein (1849–1925).

Möbius is best known for his invention of the Möbius strip. You can make a Möbius strip very easily. Simply cut a long strip of paper, twist the paper once, and connect the two ends of the strip to each other. The figure that results will look like the Möbius strip shown in Figure 4. Notice that an ant crawling along the Möbius strip will never have to pass an edge to go to "the other side." In other words, there is no "other side"; the Möbius strip has only one side.

Riemann developed some of the most important topological ideas about the stretching, bending, and twisting of surfaces. Unfortunately, he died at the early age of 39, before getting the chance to develop some of his ideas fully.

Klein is best known for the paradoxical figure illustrated in Figure 5, the Klein bottle. The Klein bottle is a one-sided object that has no edge. It consists of a tapered tube whose neck is bent around to enter the side of the bottle. The neck continues into the base of the bottle where it flares out and rounds off to form the outer surface of the bottle. Like the Möbius strip, any two points on the bottle can be joined by a continuous line without crossing an edge. This property gives the impression that the inside and outside of the Klein bottle are continuous.

[*See also* **Geometry** ]

# Topology

# Topology

Topology is sometimes called "rubber-sheet geometry" because if a shape is drawn on a rubber sheet (like a piece of a balloon), then all of the shapes you can make by stretching or twisting—but never tearing—the sheet are considered to be topologically the same.

Topological properties are based on *elastic* motions rather than rigid motions like rotations or inversions. Mathematicians are interested in the qualities of figures that remain unchanged even when they are stretched and distorted. The qualities that are unchanged by such transformations are said to be topologically invariant because they do not vary, or change, when stretched.

As an example, the figure shows a triangle, a square, a rough outline of the United States, and a ring. The first three shapes are topologically equivalent; we can stretch and pull the boundary of the square until it becomes a circle or the U.S. shape. But no matter how much we pull or stretch this basic outline we cannot make it look like a ring.

Since a triangle is topologically the same as a square, and a sphere is the same as a cone, the idea of angle, length, perimeter, area, and volume play no role in topology. What remains the same is the number of boundaries that a shape has. A triangle has an inside and an outside separated by a closed boundary line. Every possible distortion of a triangle will also have an inside and an outside separated by a boundary. The ring, on the other hand, has two boundaries forming an inside region separated from two disconnected outside regions. No matter how you transform a ring it will always have two boundaries, one inside region, and two outside regions.

It can be quite challenging and surprising to discover whether two shapes are topologically the same. For example, a soda bottle is the same as a soup bowl, which is the same as a dinner plate. A coffee cup and a donut are topologically the same. But a coffee cup is topologically different from a soup bowl because the hole in the cup's handle does not occur in the bowl.

Because topology treats shapes so differently from the way we are accustomed to thinking about them, some of the most interesting objects studied in topology may seem very strange. One of the most well known objects is called the Möbius Strip, named for the German mathematician August Ferdinand Möbius who first studied it. This curious object is a two-dimensional surface that has only one side. A Möbius Strip can be easily constructed by taking the two ends of a long, rectangular strip of paper, giving one end a half twist, and gluing the two ends together. The Klein Bottle, which theoretically results from sewing together two Möbius Strips along their single edge, is a bottle with only one side. In our three-dimensional world, it is impossible to construct because a true Klein Bottle can exist only in four dimensions.

The concepts of sidedness, boundaries, and invariants have been generalized by topologists to higher dimensions. Although difficult to visualize, topologists will talk about surfaces in four, five, and even higher dimensions. While much of the study of topology is theoretical, it has deep connections to relativity theory and modern physics which also imagine our universe as having more than three dimensions.

see also Dimensions; Mathematics, Impossible; MÖbius, August Ferdinand.

*Alan Lipp*

## Bibliography

Chinn, W.G. and Steenrod, N.E. *First Concepts of Topology.* New York: Random House, 1966.

### Internet Resources

"Beginnings of Topology." In *Math Forum.* August 98. <http://mathforum.com/~isaac/problems/bridges1.html>.

## A MUSICAL COMPARISON

A soprano and a baritone can sing the same song even though one sings high notes and the other low notes. Shifting a song up or down the musical scale changes some qualities of the music but not the pattern of notes that creates the song. Similarly, topology deals with variations that occur without changing the underlying "melody" of the shape.

# topology

to·pol·o·gy / təˈpäləjē/ •
n. 1. Math. the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures. ∎ a family of open subsets of an abstract space such that the union and the intersection of any two of them are members of the family, and that includes the space itself and the empty set.2. the way in which constituent parts are interrelated or arranged: *the topology of a computer network.*DERIVATIVES: top·o·log·i·cal / ˌtäpəˈläjikəl/ adj.top·o·log·i·cal·ly / ˌtäpəˈläjik(ə)lē/ adv.to·pol·o·gist / -jist/ n.

# topology

**topology** **1.** The study of those properties of sets that are shared by all images (homeomorphic images) of the sets under certain mappings that might be described as deformations. Topology is sometimes described as geometry done on a rubber sheet; this sheet can be pulled or stretched into different shapes. Topological properties are unaltered by distortions of this kind. Topological properties can be attributed to graphs, grammars, and even programs themselves.

**2.** (**interconnection topology**) See network architecture.

# topology

**topology** Branch of mathematics concerned with those properties of geometric figures that remain unchanged after a continuous deformation process, such as squeezing, stretching, or twisting. The number of boundaries of a surface is such a property. Any plane-closed shape (i.e. any line that eventually comes back to its beginning, all on a single plane) is topologically equivalent to a circle; a cube, a solid cone, and a solid cylinder are topologically equivalent to a sphere. See also Möbius strip

# topology

**topology** In phylogenetics, the branching pattern of a phylogenetic tree.

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