Economic

Euclidean Spaces as Metric Spaces

Dr Damien S. Eldridge

Australian National University

3 March 2021

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Reading Guide 1

Banks, J, G Elton, and J Strantzen (2009), Topology and analysis: Unit text for MAT3TA (2009 and 2010 edition), Department of Mathematics and Statistics, La Trobe University, Bundoora, February.

Corbae, D, MB Stinchcombe, and J Zeman (2009), An introduction to mathematical analysis for economic theory and econometrics, Princeton University Press, USA: Chapters 3, 4 and 6 (pp. 72-171 and 259-354).

Kolmogorov, AN, and SV Fomin (1970), Introductory real analysis (revised English edition), Translated and Edited by RA Silverman, Dover Publications, USA: Chapters 2 and 3 (pp. 37-117).

Simon, CP, and L Blume (1994), Mathematics for economists, WW Norton and Company, USA: Chapters 12 and 29 (pp. 253-272 and 803-821).

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Reading Guide 2

Spiegel, MR (1981a), Schaum’s outline of theory and problems of advanced calculus (SI metric edition), McGraw-Hill, Singapore: Chapters 2 and 3 (pp. 20-56).

Sundaram, RK (1996), A first course in optimization theory, Cambridge University Press, USA: Chapter 1, and Appendices A, B, and C.

Takayama, A (1993), Analytical methods in economics, The University of Michigan Press, USA: Chapter 1 (pp. 3-71).

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What is a Metric Space?

A metric space is a pair (X , d) that consists of a set X and a distance metric d that can be used to measure the distance between any two elements of that set.

Note that metric spaces can differ either because the underlying sets are different, or because the distance metrics are different, or for both of these reasons.

(X , d) and (Y , d) are different metric spaces if X 6= Y . (X , d) and (X , r) are different metric spaces if d 6= r. (X , d) and (Y , r) are different metric spaces if both X 6= Y and d 6= r.

We shall be particularly interested in Euclidean metric spaces. These are spaces of the form (Rn, dnE ), where d

n E is the n-dimensional

Euclidean metric and n ∈ N.

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What is a Distance Metric?

A distance metric for the set X is a function d : X ×X −→ R+ that satisfies the following three properties:

(Distinguishability): d(x, y) = 0 if and only if x = y. (Symmetry): d(x, y) = d(y, x) for all (x, y) ∈ X ×X . (Triangle Inequality): d(x, y) 6 d(x, z) + d(z, y) for all (x, y, z) ∈ X ×X ×X .

Note that “distinguishability” along with the fact that d is a “non-negative real valued” function means that d(x, y) > 0 if and only if x 6= y.

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Examples of Distance Metrics Part 1

The discrete metric over any set X :

dD(x, y) =

{ 1 if x 6= y; 0 if x = y.

The Euclidean metric over Rn:

dnE (x, y) =

{ n

∑ i=1

(yi −xi)2 }1

2

.

The sum-of-absolute-differences metric over Rn:

dnSAD(x, y) = n

∑ i=1

|yi −xi|.

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Examples of Distance Metrics Part 2

When n = 1, we have (R, dE ) = (R, dSAD) because

dE (x, y) = { (y −x)2

}1 2 = |y −x| = dSAD(x, y).

When n > 1, we have (Rn, dnE ) 6= (R n, dnSAD) because

dnE (x, y) =

{ n

∑ i=1

(yi −xi)2 }1

2

6= n

∑ i=1

|yi −xi| = dnSAD(x, y).

You will be asked to show that these three distance metrics each satisfy all of the properties of distance metrics in one of the problem sets for this course.

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Metric Sub-Spaces

Let (X , d) be a metric space and Y ⊂ X . Suppose that d|Y is the restriction of d to the set Y . In this case, the pair (Y , d|Y ) will itself be a metric space. The metric space (Y , d|Y ) is said to be a metric sub-space of the metric space (X , d).

Example: (Rn+, d n E|Rn+)is a metric subspace of (R

n, dnE ).

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Continuous mappings

Let (X , d) and (Y , r) be metric spaces.

Suppose that f : X −→ Y . The mapping f is said to be continuous at the point x0 ∈ X if, given any e > 0, there exists a δ > 0 such that r(f (x), f (x0)) < e whenever d(x, x0) < δ.

This is a formal version of the statement that limx→x0 f (x) = f (x0).

The mapping f is said to be continuous on X if it is continuous at every point x ∈ X .

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Alternative characterisations of continuity

Let (X , d) and (Y , r) be metric spaces.

Suppose that f : X −→ Y . Here are three alternative characterisations for continuity.

The mapping f is said to be continuous at the point x0 ∈ X if, given any e > 0, there exists a δ > 0 such that r(f (x), f (x0)) < e whenever d(x, x0) < δ. The mapping f is said to be continuous on X if it is continuous at every point x ∈ X . The mapping f is continuous at the point x0 ∈ X if and only if the sequence {f (xn)}∞n=1 converges to f (x0) whenever the sequence {xn}∞n=1 converges to x0. The mapping f is said to be continuous on X if it is continuous at every point x ∈ X . The mapping f is said to be continuous if f −1(A) is an open set whenever A ⊆ Y is an open set. (In other words, continuity of a mapping is equivalent to requiring that the pre-image of every open set be an open set.)

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Continuity Example 1

Consider the metric space (R, dE ) and the mapping f : R −→ R defined by f (x) = x2. We want to show that f (x) is continuous at the point x = 2.

Recall that dE (x, y) = { (y −x)2

}1 2 = |y −x|.

Note that |f (2)− f (x)| = |f (x)− f (2)| = |x2 − 4| = |x − 2||x + 2|. Suppose that |2 −x| < δ. This means that |x − 2| < δ, which in turn means that (2 − δ) < x < (2 + δ). Thus we have (4 − δ) < (x + 2) < (4 + δ). Let us restrict attention to δ ∈ (0, 1). This gives us 3 < (x + 2) < 5, so that |x + 2| < 5. We have |f (2)− f (x)| = |f (x)− f (2)| = |x − 2||x + 2| < (δ)(5) = 5δ. We require |f (2)− f (x)| < e. One way to ensure this is to set δ = min(1, e

5 ).

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Continuity Example 2

Consider the metric space (R, dE ) and the mapping f : R −→ R defined by f (x) = x. We want to show that f (x) is continuous for all x ∈ X . Recall that dE (x, y) =

{ (y −x)2

}1 2 = |y −x|.

Consider an arbitrary point x = a. Note that |f (a)− f (x)| = |a−x|. Suppose that |a−x| < δ. Note that if we set δ = e , we have |f (a)− f (x)| < e. Thus we know that f is continuous at the point x = a.

Since a was chosen arbitrarily, we now that this is true for all a ∈ R. This means that f is continuous for all x ∈ R.

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Continuity Example 3: Part 1

Consider the metric spaces (R, dE ) and (R 2, d2E ), along with the

mapping f : R2 −→ R defined by

f (x1, x2) = x 2 1 + x

2 2 .

We want to show that f (x1, x2) is continuous at the point (x1, x2) = (0, 0).

Recall that

d2E (x, y) =

{ 2

∑ i=1

(yi −xi)2 }1

2

and

dE (x, y) = { (y −x)2

}1 2 = |y −x|.

Note that

d2E ((x1, x2), (0, 0)) = { (0 −x1)2 + (0 −x2)2

}1 2 =

{ x21 + x

2 2

}1 2 .

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Continuity Example 3: Part 2

Thus d2E ((x1, x2), (0, 0)) < δ requires that { x21 + x

2 2

}1 2 < δ. This in

turn requires that x21 + x 2 2 < δ

2.

We have

d(f (x1, x2), f (0, 0)) = |(x21 + x 2 2 )−(0

2 + 02)| = |x21 + x 2 2 | = x

2 1 + x

2 2 .

We require that

d(f (x1, x2), f (0, 0)) = x 2 1 + x

2 2 < e.

This can be achieved by setting δ = √

e.

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Discontinuity Example 1: Part 1

Consider the metric space (R, dE ) and the mapping f : R −→ R defined by

f (x) =

{ 1 if x > 5;

0 if x < 5.

We want to show that f (x) is not continuous at the point x = 5.

Recall that dE (x, y) = { (y −x)2

}1 2 = |y −x|.

Note that

dE (f (x), f (5)) = |f (5)− f (x)| = |1 − f (x)| = {

0 if x > 5;

1 if x < 5.

If |5 −x| < δ, then 5 − δ < x < 5 + δ.

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Discontinuity Example 1: Part 2

Consider some point x0 ∈ (5 − δ, 5) ⊂ (5 − δ, 5 + δ). Since δ > 0, we know that (5 − δ, 5) 6= ∅. Suppose that e = 1

2 .

In order for dE (f (x), f (5)) = 1 − f (x) < 12 , we would need δ 6 0. Thus there does not exist any δ > 0 that will ensure that dE (f (x), f (5)) <

1 2

.

This means that f is not continuous at the point x = 5.

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Discontinuity Example 2

Consider the metric space (R, dE ) and the mapping f : R −→ R defined by

f (x) =

{ 1 x

if x 6= 0; 0 if x = 0.

This function is a rectangular hyperbola when x 6= 0, but it takes on the value 0 when x = 0.

Recall that the rectangular hyperbola part of this function is not defined at the point x = 0.

This function is discontinuous at the point x = 0.

Illustrate the graph of this function on the whiteboard.

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Discontinuity Example 3

Consider the metric space (R, dE ) and the mapping f : R −→ R defined by

f (x) =

{ 1 if x ∈ Q; 0 if x /∈ Q.

This function is sometimes known as Dirichlet’s discontinuous function.

It is discontinuous at every point in its domain.

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Open and Closed Balls

Let (X , d) be a metric space.

An open ball with radius e around the point x0 ∈ X in this metric space is a set of the form

B(x0, e) = {x ∈ X : d(x, x0) < e} .

This set is sometimes known as an e-neighbourhood of x0.

A closed ball with radius e around the point x0 ∈ X in this metric space is a set of the form

B(x0, e) = {x ∈ X : d(x, x0) 6 e} .

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Economic Application: Local Non-Satiation

Suppose that (X , d) is a metric space, % is a rational weak preference relation defined on X , and � is the induced strict preference relation on X .

% is said to be locally non-satiated at the point x ∈ X if, for each e > 0, there exists some other point x′ ∈ Be(x) such that x′ � x.

Note that that Be(x) = {y ∈ X : d(y, x) < e}. % is said to be locally non-satiated on X if it is locally non-satiated at every point x ∈ X . Local non-satiation of preferences on X ensures that:

Individuals exhaust their budgets; Bliss points do not exist; and Indifference curves are not fat.

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Contact Points and Set Closure

Let (X , d) be a metric space and Y ⊆ X . A point x ∈ X is said to be a contact point of Y if every neighbourhood of x contains at least one point in Y .

Formally, this can be stated as follows.

A point x ∈ X is a contact point of Y if B(x, e)∩Y 6= ∅ for all e > 0. The closure of Y , which is denoted by [Y ], is the set of all contact points of Y .

Formally, this can be stated as follows.

[Y ] = {x ∈ X : B(x, e)∩Y 6= ∅ for all e > 0}. Note that Y ⊆ [Y ] because, for any e > 0, we have B(y, e)∩Y 6= ∅ for all y ∈ Y . Illustrate contact points and set closure on the whiteboard.

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Some Properties of Set Closure

If Y ⊆ X , then [Y ] ⊆ [X ]. [[X ]] = [X ].

[X ∪Y ] = [X ]∪ [Y ]. [∅] = ∅. Proofs for the last three of these properties can be found in Kolmogorov and Fomin (1970, pp. 46-47). According to them, the first property is obvious.

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The Boundary of a Set in a Metric Space

Let (X , d) be a metric space and consider a set A ⊂ X . The complement of the set A (with respect to X ) is AC = X \A. The closure of A is [A].

The closure of AC is [ AC ] = [X \A].

The boundary of the set A is ∂A = [A]∩ [ AC ] .

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Limit Points and Isolated Points

Let (X , d) be a metric space and Y ⊆ X . A point x ∈ X is said to be a limit point of Y if every neighbourhood of x contains infinitely many points in Y .

The point x can be an element of Y , but it does not have to be an element of Y . For example, suppose that Y = {y ∈ Q : y ∈ [0, 1]}. (That is, Y is the set of rational numbers in the [0, 1] interval.) In this case, every point in [0, 1] interval, both rational and irrational, is a limit point of Y .

A point y ∈ Y is said to be an isolated point of Y if there is some neighbourhood of y containing no points of Y other than itself.

Formally, this can be stated as follows. A point y ∈ Y is an isolated point of Y if there exists some e > 0 such that B(y, e)∩Y = {y}.

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Sequences

A sequence from a set X is a mapping of the form f : N −→ X . In effect, it is an assignment of an element from X to each of the natural numbers. It is often denoted by {xn}∞n=1 or {xn}n∈N, where xn ∈ X for all n ∈ N. It looks like {x1, x2, · · · , xn, · · ·}.

Note that this definition of a sequence requires that all sequences be of countably infinite length.

It does not explicitly allow for finite sequences. But we can think about a finite sequence as being a truncated sequence, where we throw away all terms for which n > n̂ for some n̂ ∈ N. Such a sequence would be written as {xn}n̂n=1. In practice, we are often interested in the limiting behaviour of sequences, so that the restriction to infinite sequences is not a problem.

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Sub-Sequences

Let g : N −→ N be a strictly increasing map of the form g(k) = nk .

This means that g(k + 1) = nk+1 > nk = g(k) for all k ∈ N. If the map f : N −→ X is a sequence from X , then so is the map h : N −→ X given by h = f ◦g.

Note that h(k) = f (g(k)) = f (nk). If f generates the sequence {x1, x2, · · · , xn1−1, xn1 , xn1+1, · · · , xn2−1, xn2 , xn2+1, · · ·}, then h generates the sequence {x1, x2, · · · , xn1 , · · · , xn2 , · · ·}.

The sequence {xnk}nk∈N is a sub-sequence of the sequence {xn}n∈N. Note that every term in the sequence {xnk}nk∈N falls somewhere in the sequence {xn}n∈N. Note also that the (relative) order in which the terms appear is the same for each of the sequences.

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Sub-Sequence Example 1

Consider the sequence {xn}n∈N where xn = 1 n

.

The map for this sequence is f (n) = 1n .

The sequence looks like {

1, 12 , 1 3 , · · · ,

1 n , · · ·

} .

Consider the strictly increasing map g : N −→ N defined by g(k) = k2.

This , along with f , generates the sub-sequence map

h(k) = f (g(k)) = f (k2) = 1

k2 .

The associated sub-sequence is{ 1

k2

} k∈N

=

{ 1,

1

4 ,

1

9 , · · · ,

1

n2 , · · ·

} .

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Sub-Sequence Example 2

Consider the sequence {xn}n∈N where xn = 1 n

.

The map for this sequence is f (n) = 1n .

The sequence looks like {

1, 12 , 1 3 , · · · ,

1 n , · · ·

} .

Consider the strictly increasing map g : N −→ N defined by g(k) = 2k .

This , along with f , generates the sub-sequence map

h(k) = f (g(k)) = f (2k) = 1

2k .

The associated sub-sequence is{ 1

2k

} k∈N

=

{ 1

2 ,

1

4 ,

1

8 , · · · ,

1

2n , · · ·

} .

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Convergence and Limits

Let (X , d) be a metric space, {xn}n∈N be a sequence from X and x ∈ X be a point in X . The sequence {xn}n∈N is said to be a convergent sequence if, for each e > 0, there exists n̂e ∈ N such that d(xi , xj) < e for all (i, j) ∈{(k, l) ∈ N × N : k > n̂e, l > n̂e}.

A convergent sequence is known as a Cauchy sequence.

The sequence {xn}n∈N is said to converge to x ∈ X if, for each e > 0, there exists n̂e ∈ N such that xn ∈ B(x, e) for all n > n̂e.

In this case, we say that x is the limit of the sequence {xn}n∈N. If a sequence converges to a particular limit, say x, then so does every sub-sequence of that sequence.

Sequences in metric spaces can have at most one limit.

Such a sequence either converges to a unique limit or it does not converge. This is not true for sequences in some non-metrisable topological spaces. However, we will not encounter such spaces in this subject.

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Convergence Examples Part 1

These example relate to the metric space (R, dE ).

The sequence {

1 n

} n∈N

converges to the point x = 0.

The sequence {

1 n2

} n∈N

converges to the point x = 0.

Recall that {

1 n2

} n∈N is a sub-sequence of

{ 1 n

} n∈N.

The sequence {

1 2n

} n∈N

converges to the point x = 0.

Recall that {

1 2n } n∈N is a sub-sequence of

{ 1 n

} n∈N.

The sequence {

(−1)n n

} n∈N

= { −1, 12 ,

−1 3 , · · ·

} converges to the

point x = 0.

The sequence {

n n+1

} n∈N =

{ 1 2 ,

2 3 ,

3 4 , · · ·

} converges to the point

x = 1. The sequence {(−1)n}n∈N = {−1, 1,−1, 1, · · ·} does not converge. The sequence {n}n∈N does not converge.

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Convergence Examples Part 2

These example relate to the metric space (R2, d2E ).

The sequence { (1n ,

(n−1) n )

} n∈N

converges to (0, 1).

The sequence { ( (−1)n

n , (−1)n

n ) } n∈N

converges to (0, 0).

The sequence {((−1)n, (−1)n)}n∈N does not converge. The sequence {(n, n)}n∈N does not converge.

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Convergence in Euclidean spaces

Suppose {xk}k∈N is a sequence of points in (R n, dnE ).

Note that xk = (x1,k , x2,k , · · · , xn,k), where xi,k ∈ R for all i ∈{1, 2, · · · , n}.

The sequence {xk}k∈N converges in (R n, dnE ) if and only if each of

the n component sequences of the form {xi,k}k∈N converge in (R, dE ).

This is Theorem 12.5 in Simon and Blume (1994, p. 262). A proof can be found in Simon and Blume (pp. 262-263).

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Economic Application: Continuity of Preferences

Suppose that (X , d) is a metric space and % is a rational weak preference relation defined on X .

Let {xn}n∈N and {yn}n∈N be a pair of convergent sequences from X such that:

xn % yn for all n ∈ N, limn→∞ xn = x∞ ∈ X , and limn→∞ yn = y∞ ∈ X .

The weak preference relation % is said to be continuous if, for any such pair of sequences, we have x∞ % y∞.

We will explore this application further in one of the problem sets for this course.

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Economic Application: Equilibria in Games

Some equilibrium concepts in non-cooperative games make use of convergent sequences of beliefs.

These include:

Sequential equilibria in extensive form games; Trembling-hand perfect equilibria in extensive form games; Trembling-hand perfect equilibria in normal form games; and Proper equilibria in normal form games.

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Econometric Application: Asymptotic Results

Various asymptotic results in econometrics make use of convergent sequences.

Examples include the following. The consistency of some estimators.

This involves convergence in probability. An estimator θ̂ of a parameter θ is said to be consistent if plim θ̂ = θ.

The asymptotic distribution of some transformation of an estimator.

This involves the convergence in distribution of some transformation of an estimator. A transformation is typically needed to stabilise the variance of the asymptotic distribution (that is, prevent the asymptotic variance of the estimator from collapsing to zero). For example, sometimes we will get something like √ n(θ̂ − θ) d−→ N(0, σ2).

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Dense subsets

Consider a metric space (X , d) and two sets, A ⊆ X and B ⊆ X . The set A is said to be dense in B if B ⊆ [A]. The set A is said to everywhere dense (meaning dense in X ) if [A] = X .

Note that [A] = X if and only if both X ⊆ A and [A] ⊆ X . The set A is said to be nowhere dense if Be(x) * [A] for any (x, e) ∈ X × R+. Example: Q is everywhere dense in R because [Q] = R.

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Separable metric spaces

A metric space (X , d) is said to be a separable metric space if there exists A ⊆ X such that A is both countable and everywhere dense. Example 1: (R, dE ) is separable because Q is both countable and everywhere dense in R.

Example 2: (Rn, dnE ) is separable because Q n is both countable and

everywhere dense in Rn.

Example 3: (X , dD), where dD is the discrete metric, is separable if and only if X is a countable set. (Note that X is everywhere dense in X under the discrete metric because [X ] = X under the discrete metric).

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Closed sets

Consider a metric space (X , d). Let A ⊆ X . A is a closed set if A = [A].

In other words, a closed set is one that contains all of its limit points.

Intuitively, a closed set is one that includes all of its boundary points.

Examples: ∅, X , Be(x) for any x ∈ X , and any finite set. Theorem: The intersection of an arbitrary number of closed sets is itself a closed set.

Theorem: The union of a finite number of closed sets is itself a closed set.

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Open sets

Consider a metric space (X , d). Let A ⊆ X . x ∈ A is said to be an interior point of A if there exists some e > 0 such that Be(x) ⊆ A. A is an open set if all of its elements are interior points.

Intuitively, an open set is a set that includes none of its boundary points.

Examples:

(a, b) ⊆ R where a < b and (X , d) = (R, dE ). Be(x) for any (e > 0 and x ∈ X ).

Theorem: A ⊆ X is open if and only if X�A is closed. Corrollary: X and ∅ are open.

Theorem: The union of an arbitrary number of open sets is itself an open set.

Theorem: The intersection of a finite number of open sets is itself an open set.

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Clopen Sets

Let (X , d) be a metric space.

Some subsets of X are both open and closed. Such sets are called clopen sets.

Example 1: X is clopen.

X is open because Be(x) ⊆ X for all (x, e) ∈ X × R+ by definition. X is closed because X = X�∅ and ∅ is open.

Example 2: ∅ is clopen. ∅ is open because Be(x) ⊆ ∅ for all (x, e) ∈ ∅× R+ by definition. ∅ is closed because ∅ = X�X and X is open.

Example 3: If X is a finite set, then every element of 2X is clopen.

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Sets that are neither open nor closed

Let (X , d) be a metric space.

It is possible that there are some subsets of X that neither open nor closed.

Such sets will include some of their boundary points but not all of them.

Example 1: [a, b) ⊆ R where a < b in the metric space (R, dE ). Example 2: (a, b] ⊆ R where a < b in the metric space (R, dE ).

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Open sets of real numbers

The structure of open sets in a metric space can often be quite complicated. This is true even for Euclidean spaces with two or more dimensions (that is, for metric spaces of the form (Rn, dnE ) where n > 2).

However, it is relatively straightforward to characterise all of the open sets in one dimensional Euclidean space (that is, for the metric space (R, dE )).

Every open set on the real line is the union of either a finite or countably infinite number of pairwise disjoint open intervals.

This requires that (−∞, ∞), (−∞, a), (a, ∞) and (a, a) = ∅ be considered to be open intervals on the real line.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 42 / 56

Closed sets of real numbers

The structure of closed sets in a metric space can often be quite complicated. This is true even for Euclidean spaces with two or more dimensions (that is, for metric spaces of the form (Rn, dnE ) where n > 2).

However, it is relatively straightforward to characterise all of the closed sets in one dimensional Euclidean space (that is, for the metric space (R, dE )).

Every closed set on the real line can be obtained by deleting either a finite or countably infinite set of pairwise disjoint open intervals from the real line.

This requires that (−∞, ∞), (−∞, a), (a, ∞) and (a, a) = ∅ be considered to be open intervals on the real line.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 43 / 56

Various Equivalent Definitions of Set Closure

This is Theorem 4.5.6 in Corbae, Stinchcombe, and Zeman (2009).

Let (X , d) be a metric space and consider a set A ⊆ X . The following statements are equivalent.

1 A is closed. 2 A = [A]. 3 A = ∩{Ae : e > 0}, where Ae = ∪a∈AB (a, e) is the open epsilon-ball

around the set A.

Corbae, Stinchcombe, and Zeman suggest that this version of the definition of set closure might be accurately described as “the shrink-wrap definition of set closure”. Can you explain why they might hold this view?

4 A = {x ∈ X : ∀e > 0, B (x, e)∩A 6= ∅}. 5 A =

{ x ∈ X : ∃

[ {an}n∈N , an ∈ A ∀n ∈ N

] , an −→ x

} .

6 ∂A ⊆ A.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 44 / 56

Complete metric spaces

Let (X , d) be a metric space and {xn}n∈N be a Cauchy sequence from X .

Recall that a convergent sequence is called a Cauchy sequence.

Suppose that the limit of the Cauchy sequence {xn}n∈N is the point x̂.

It is possible that x̂ /∈ X . For example, a Cauchy sequence from Q might converge to an irrational number like

√ 2.

If any Cauchy sequence from a metric space converges to a point outside that space, then the metric space is said to be incomplete.

A metric space (X , d) is said to be a complete metric space if every Cauchy sequence from that space converges to an element of that space.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 45 / 56

The Nested Sphere Theorem

Let (X , d) be a metric space and { B(xn, en)

} n∈N be a sequence of

closed balls in X .

Recall that a closed ball (or closed sphere) centred on the point x ∈ X with radius e > 0 is a set of the form

B(x, e) = {y ∈ X : d(y, x) 6 e} .

The sequence { B(xn, en)

} n∈N is said to be nested (or decreasing) if

B(xk+1, ek+1) ⊆ B(xk , ek) for all k ∈ N.

Theorem: A metric space (X , d) is complete if and only if every nested sequence

{ B(xn, en)

} n∈N of closed balls in X for which

en −→ 0 as n −→ ∞ has a non-empty intersection ∩n∈NB(xn, en).

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 46 / 56

Baire’s Theorem

Let (X , d) be a metric space and {An}n∈N be a sequence of nowhere dense subsets of X .

Recall that a set An ⊆ X is nowhere dense in X if it is not dense in any open ball in X . That is, a set An ⊆ X is nowhere dense in X if [An] 6= B(x, e) for any (x, e) ∈ X × R+.

Theorem: A complete metric space (X , d) cannot be represented as the union of a countable number of nowhere dense subsets of the space.

Corollary: A complete metric space (X , d) without any isolated points is uncountable.

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The metric completion theorem

An incomplete metric space can always be enlarged (in an essentially unique way) to form a complete metric space.

Suppose that (X , d) is an incomplete metric space.

The metric space (X∗, d∗) is called the completion of (X , d) if X ⊂ X∗ and [X ] = X∗.

That is, (X∗, d∗) is called the completion of (X , d) if X ⊂ X∗ is everywhere dense in X∗. The distance metric d∗ is constructed by extending the domain of the distance metric d to include all elements of X∗�X , as well as all of the elements of X itself.

Theorem: Every incomplete metric space (X , d) has a completion. This completion is unique to within an isometric mapping carrying every point x ∈ X into itself. Example: (R, dE ) is the completion of (Q, dE|Q).

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Compact Metric Spaces and Compact Sets Part 1

Let (X , d) be a metric space, I be an index set and C = {Ai}i∈I be a collection of subsets of X .

C is called a cover for X if X = ∪i∈I Ai . C is called an open cover for X if both C is a cover for X and Ai is an open set for all i ∈ I . C is called a closed cover for X if both C is a cover for X and Ai is a closed set for all i ∈ I . If S ⊂C is also a cover for X , then S is called a sub-cover of C. A metric space (X , d) is said to be a compact metric space if every open cover of X has a finite sub-cover.

This is known as the Heine-Borel Property. Do not confuse the Heine-Borel Property with the (related) Heine-Borel Theorem that we will encounter shortly.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 49 / 56

Compact Metric Spaces and Compact Sets Part 2

Theorem: If (X , d) is a compact metric space and Y ⊆ X is a closed set, then (Y , d|Y ) is a compact metric sub-space of (X , d). If (X , d) is a compact metric space and (Y , d|Y ) is a compact metric sub-space of (X , d), we will often refer to X and Y as being compact sets in the metric space (X , d).

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 50 / 56

Continuous Mappings with a Compact Domain Part 1

Theorem: If (X , d) is a compact metric space, (Y , r) is any metric space, and f : X −→ Y is a continuous mapping of Y onto X , then Y = f (X) is itself compact.

The proof follows below.

This proof comes from Kolmogorov and Fomin (1970, p. 94). Let {Vi}i∈I be any open cover of Y . This means that Vi is open for all i ∈ I and ∪i∈I Vi = Y . Since f is a continuous mapping and Vi is open for all i ∈ I , we know that f −1(Vi ) is open for all i ∈ I . Since f is onto and ∪i∈I Vi = Y , we know that f −1(∪i∈I Vi ) = X . Recall that f −1(∪i∈I Vi ) = ∪i∈I f −1(Vi ). This means that ∪i∈I f −1(Vi ) = X . Thus {Ui}i∈I =

{ f −1(Vi )

} i∈I is an open cover for X .

Proof continued on next slide.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 51 / 56

Continuous Mappings with a Compact Domain Part 2

Proof continued from previous slide.

Since X is compact, we know that there exists some finite sub-cover{ Uxj

} j∈{1,2,··· ,n}

⊂{Ui}i∈I . This means that ∪j∈{1,2,··· ,n}Uxj = X . Since f is onto, we know that f (∪j∈{1,2,··· ,n}Uxj ) = Y . Recall that f (∪j∈{1,2,··· ,n}Uxj ) = ∪j∈{1,2,··· ,n}f (Uxj ). Thus we have ∪j∈{1,2,··· ,n}f (Uxj ) = Y , which means that{ f (Uxj )

} j∈{1,2,··· ,n}

is a finite cover for Y .

Since { Uxj

} j∈{1,2,··· ,n}

⊂{Ui}i∈I , we know that{ f (Uxj )

} j∈{1,2,··· ,n}

⊂{Vi}i∈I .

This means that { f (Uxj )

} j∈{1,2,··· ,n}

is a finite sub-cover for the open

cover {Vi}i∈I of Y . Thus the set Y is compact.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 52 / 56

Boundedness

Let (X , d) be a metric space and Y ⊆ X . The diameter of the set Y is defined to be D(Y ) = sup(x,y)∈Y×Y d(x, y). The set Y is said to be bounded if its diameter is finite.

In other words, the set Y is bounded if there exists some finite e > 0 such that d(x, y) < e for all (x, y) ∈ Y ×Y .

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Total Boundedness

Let (X , d) be a metric space and e > 0 be any strictly positive real number. Then A ⊆ X is said to be an e-net for Y ⊆ X if, for every x ∈ Y , there is at least one point a ∈ A such that d(x, a) 6 e. LetX , d) be a metric space and Y ⊆ X . The set Y is said to be totally bounded if it has a finite e-net for every e > 0.

Note that if Y is totally bounded, then so is [Y ]. In Euclidean n-space, (Rn, dnE ), total boundedness is equivalent to boundedness.

A metric space (X , d) is compact if and only if it is both totally bounded and complete.

A subset Y of a complete metric space (X , d) is relatively compact if and only if it is totally bounded.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 54 / 56

Compactness in Euclidean metric spaces

If (Rn, dnE ) is the metric space under consideration and Y ⊆ X , then Y is a compact set if and only if it is both closed and bounded.

This is only true for Euclidean metric spaces. It is not true for other metric spaces and other topological spaces. This is known as the Heine-Borel Theorem. Do not confuse this with the Heine-Borel property that we encountered earlier.

Example 1: If prices and income are strictly positive and finite, then

the Marshallian budget set, { x ∈ RL+ : ∑

L l=1 plxl 6 y

} , is compact.

Example 2: If prices and income are strictly positive and finite, then

the Walrasian budget set, { x ∈ RL+ : ∑

L l=1 pl(xl −el) 6 0

} , is

compact.

D. S. Eldridge (ANU) Euclidean Metric Spaces 3 March 2021 55 / 56

Why does compactness matter?

Finite sets have many desirable properties that are not possessed by infinite sets in general.

Infinite sets that are compact possess some of these desirable properties.

An important application of compactness and continuity in economics concerns solutions to optimisation problems in general and constrained optimisation problems in particular.

If the objective function is continuous and the constraint set is compact, then that function will always achieve a maximum value and a minimium value on that set.

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