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Topology
Entry prepared for
The International Encyclopedia of the Social Sciences
First version: December 2005
This version: September 2006
Topology is a mathematical structure designed to express the ideas of robustness,
approximation, convergence and continuity. In the social sciences, we often like to
know if close-by models of human behavior and interaction entail close-by predictions.
More precisely, we would like to know if 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.
For example, di erent economies di er by the endowments of individuals, by the
production technologies available to them, and by their preferences over bundles of
commodities. General Equilibrium theory provides a prediction of the market prices
that can emerge for each such combination. Considering a topological space of economies
enables one to examine whether market prices changes continuously with the economies'
characteristics.
Similarly, in Game Theory, Nash equilibrium is a prediction of the strategies that the
players will choose non-cooperatively as a function of their preferences over the outcomes
entailed by strategy pro les. When considering a topological space of games, one can
check how Nash equilibria vary with the game speci cation. Alternatively, Social Choice
Theory, based on normative considerations, prescribes a strategy pro le that the players
should choose collectively (rather than non-cooperatively) 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
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of nitely many 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 de nes the distance between any two objects in the space1 . In such a case,
the unions of open balls2 constitute a topology. Moreover, a set V is open if and only if
every point x 2 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 2 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, that 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 xn 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 xn , n N in the sequence
belong to V .
This de nition of convergence applies not only to sequences but also to nets. In a
net xn , the indices n are not necessarily the natural numbers. Rather, they may form a
directed system { a set where not in every pair of distinct indices n; n0 , one of the indices
is necessarily larger than the other, but there always exists another index n00 which 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 xn converges to x only if for some index N and onwards xn = x for all n
N.
Hence, the choice of topology expresses the extent to which the modeler views di erent
points (or objects or model characteristics) in the space as distinct or similar. The more
ne-detailed the distinctions are, the richer will be the topology, and fewer nets will be
converging to any given point x.
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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.
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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.
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When X and Y are topological spaces, we say that the function f : X ! Y is
continuous at the point x 2 X if for every net xn converging to x, the net f (xn ) converges
to the point f (x) 2 Y . We say that f is continuous if it is continuous at every point
x 2 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 mis-speci cation 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
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f (W ) = fx 2 X : f (x) 2 W g is open in X. Thus, the richer is the topology of X and
the poorer is the topology of Y , more functions f from X to Y are continuous.
References
The literature on topology is vast. Accessible introductions can be found e.g. in
Aliprantis, Charalambos D. and Kim C. Border. 2006. In nite Dimensional Analysis {
A Hitchhiker's Guide, 3nd ed. Springer.
Royden, Halsey. 1988. Real Analysis, 3rd ed. Prentice-Hall (4th edition expected 2007)
Aviad Heifetz
The Economics and Management Department
The Open University of Israel
http://www.openu.ac.il/Personal sites/Aviad-Heifetz.html
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