Competitive Equilibrium: Theory and Applications

Bryan Ellickson

Chapter 4 Topology - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that:
(a) $T_4 \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1$ (so that, for example, a topology which is $T_3$ is automatically $T_2$ and $T_1$ ).
(b) If $X$ has the discrete topology, then it is $T_4$.
(c) If $X$ has the indiscrete topology and contains more than one point, then it is not $T_1$. Verify that in this case one point sets are not closed.

Brandon Collins

Numerade Educator

03:55

Problem 2

Verify that the collection of open intervals in $\mathbf{R}$ meets the requirements for a topological basis.

Brandon Collins

Numerade Educator

Problem 3

(a) Prove that $\bigcap_{\alpha \in R_{+}}[-\alpha, \alpha]$ is a closed set.
(b) Is $\bigcap_{a \in \mathbf{R}_{++}}(-\alpha, \alpha)$ an open set? A $G_\delta$ set? A Borel set?

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Problem 4

Show that a subbasis for the product topology is given by the sets of the form $\prod_{\alpha \in A} B_\alpha$ where exactly one of the $B_\alpha$ is a subbasis element for the topology on $X_\alpha$ and $B_\alpha=X_\alpha$ for all of the other $\alpha \in A$. (Hint: All that you need to show is that a typical basis element can be expressed as a finite intersection of these subbasis elements.) Illustrate the procedure by showing how an arbitrary open rectangle in $\mathbf{R}^2$ can be expressed as the intersection of two subbasis elements of this form.

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Problem 5

Let $\mathbf{Q}$ denote the set of rational numbers in $\mathbf{R}$ (i.e., those real numbers which are expressible as the ratio of two integers) and let $\mathbf{Q}_{++}$ be the set of rationals which are strictly positive. Show that for any metric space $(X, \rho)$, the collection of balls centered at $x$ with radius $\epsilon \in Q_{++}$-onstitntes a basis for the metric topology (and, hence, conclude that every metric space is first countable). Observe that you could even restrict the collection of basis elements at $x$ to be those balls with radius $1 / n$ where $n \in \mathbf{Z}_{++}$.

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01:16

Problem 6

Let $\mathcal{S}$ be a collection of subsets of a set $X$.
(a) Verify that $\subset$ is a partial ordering of $\mathcal{S}$.
(b) Give an example of a collection $S$ for which $\subset$ is not a total ordering; an example where $C$ is a total ordering.

Heather Zimmers

Numerade Educator

Problem 7

Since the discrete topology is the finest possible topology on a set, convergence relative to this topology should be very strong. At the opposite extreme, since the indiscrete topology is the coarsest topology on a set, convergence should be very weak.
(a) Prove that if $X$ has the discrete topology then a net $x^\alpha \rightarrow x$ iff there exists an $\alpha^*$ such that $x^\alpha=x$ for all $\alpha \succeq \alpha^*$.
(b) Show that if $X$ has the indiscrete topology then any net in $X$ converges to every point in $X$ !

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03:55

Problem 8

Let $S$ be an arbitrary subset of a topological space $(X, \tau)$.
(a) Prove that the set int $S$ is open and the set $\mathrm{cl} S$ is closed.
(b) Prove that (i) $S$ is closed iff $S=\operatorname{cl} S$ and (ii) $S$ is open iff $S=\operatorname{int} S$
(c) Show that $\mathrm{el}=S \cup \operatorname{acc} S$.

Brandon Collins

Numerade Educator

Problem 9

Suppose that the point $x$ in a topological space $(X, \tau)$ is an isolated point as defined above: i.e., $x$ is not accessible as the limit of a net of points in $X$ other than $x$ itself.
(a) Show that $\{x\}$ is both open and closed.
(b) Let $S$ be a set containing $x$. Show that $x \notin$ acc $S$.

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01:00

Problem 10

Draw some pictures of sets in $\mathbf{R}^2$ which are neither open nor closed. Describe the interior, closure, boundary, and set of accumulation points for each of the sets you have drawn.

Raj Bala

Numerade Educator

Problem 11

Using the fact that every open interval contains a rational number, prove that $\mathbf{Q}$ is a dense subset of $\mathbf{R}$ under the standard topology.

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Problem 12

Show that a fuuction $f: \mathbf{R} \rightarrow \mathbf{R}$ is continuous iff the sets $f^{-1}(-\infty, a)$ and $f^{-1}[a,-\infty)$ are closed in $\mathbf{R}$ for every $a \in \mathbf{R}$. Verify that the function $f: \mathbf{R} \rightarrow \mathbf{R}, f: x \mapsto x^2$ is continuous.

Ma. Theresa Alin

Numerade Educator

Problem 13

Let $X, Y$, and $Z$ be topological spaces and assume that the functions $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are continuous. Prove that the composite function $f \circ g: X \rightarrow Z$ is continuous. (Hint: First show that $f^{-1}\left(g^{-1}(G)\right)=(g \circ f)^{-1}(G)$ for any $G \subset Z$ whether $G$ is open or not.)

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Problem 14

(a) A function $f: X \rightarrow Y$ is called an open map if it maps every open set in $X$ to an open set in $Y$ and a closed map if it maps every closed set in $X$ to a closed set in $Y$. Provide an example of a function $f: \mathbf{R} \rightarrow \mathbf{R}$ which is continuous but not open and one which is continuous but not closed.
(b) Prove that if a function $f: X \rightarrow Y$ is bijective and open, then it is also closed, and that if it is bijective and closed it must also be open.

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01:01

Problem 15

Let $S_2$ and $S_2$ be any two subsets of $\mathbf{R}^2$ whose closures in $\mathbf{R}^2$ are disjoint. Prove that $Y=S_1 \cup S_2$ is disconnected even though $S_1$ and $S_2$ are not necessarily open in the topology of $\mathbf{R}^2$.

Raj Bala

Numerade Educator

03:55

Problem 16

Prove that if $(X, \tau)$ is a topological space with the discrete topology, then $X$ is compact iff it contains a finite number of elements.

Brandon Collins

Numerade Educator

05:38

Problem 17

Prove DeMorgan's Law.

Amany Waheeb

Numerade Educator

Problem 18

Let $f: X \rightarrow Y$ be a continuous function from a topological space $X$ to a topological space $Y$.
(a) Prove that the image $f(C)$ of a connected set $C$ is connected and that the image $f(K)$ of a compact set $K$ is compact.
(b) Using a continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$, provide examples illustrating that the inverse image $f^{-1}(C)$ of a connected set $C$ need not be connected and that the inverse image $f^{-1}(K)$ of a compact set need not be compact.

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Problem 19

(a) Show that the Cartesian product of compact topological spaces is compact if the product is given any topology weaker than the product topology.
(b) Show that, unless all but a finite number of the coordinate spaces have the trivial topology, the Tychonoff Theorem fails if the box topology is used in place of the product topology and the index set $A$ is not finite. (Hint: For each $\alpha \in A$ let $B_\alpha^*$ be an open proper subset of $X_\alpha$ and consider sets of the form $\tilde{B}_{\alpha^{\prime}}=\prod_{\alpha \in A} B_\alpha$ where $B_\alpha=B_\alpha^*$ for all but one $\alpha\left(=\alpha^{\prime}\right)$ for which $B_\alpha=X_\alpha$. Show that the collection $\left\{\vec{B}_\alpha \mid \alpha \in A\right\}$ is an open cover of $X=\prod_{\alpha \in A} X_\alpha$ with the box topology and that this cover has no finite subcover.)

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07:00

Problem 20

As noted in Section 4.2.1, any linear topology can be completely described in terms of a local base at 0. It is tempting to conclude that this proves that any first countable TVS is immediately second countable, but this is false. Why?

Oswaldo Jiménez

Numerade Educator

04:27

Problem 21

Illustrate translation invariance in $\mathbf{R}^2$ using the Euclidean metric; the sup metric. Verify that the basis elements for each of these metric topologies in $\mathbf{R}^{\mathbf{n}}$ are convex sets.

Anthony Ramos

Numerade Educator

02:01

Problem 22

A subset of a vector space $L$ is called a line if it is a one-dimensional subspace or a translate of a one-dimensional subspace. Let $A \subset L$ and suppose that $a \in A$. Prove that $a \in \operatorname{lin} A$ if and only if every line $S$ containing a contains a linearly open interval $(x, y) \subset A \cap S$ such that $a \in(x, y)$.

Harshita Goel

Numerade Educator

04:16

Problem 23

Give an example of two convex sets $A, B \in \mathrm{R}^2$ for which $\mathrm{cl} A+\mathrm{cl} B \subset$ $\operatorname{cl}(A+B)$ but $\operatorname{cl} A+\operatorname{cl} B \neq \operatorname{cl}(A+B)$. (Hint: Since $A$ and $B$ are convex, you may find it convenient to replace $\operatorname{cl} A, \mathrm{cl} B$, and $\operatorname{cl}(A+B)$ by their equivalent linear counterparts: $\operatorname{lcl} A, \operatorname{lcl} B$, and $\operatorname{lcl}(A+B)$.)

Chris Trentman

Numerade Educator

Problem 24

Prove Theorem 4.40.

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Problem 25

Show that the linear functional $0 \in L^{\prime}$ belongs to $L^*$ whatever the topology on $L$. (Hint: Note that the inverse image of any open set containing the point 0 equals $L$ and the inverse image of any open set not containing 0 is empty.)

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01:54

Problem 26

Give an example of two disjoint, closed, convex subsets of $\mathbf{R}^2$ which can be separated but not strictly or strongly separated by a hyperplane. (Hint: One of the sets must be noncompact.)

Chris Trentman

Numerade Educator

04:16

Problem 27

Show that if $A$ is a nonempty, convex, and open subset of a TVS, then it has no support points. What are the support points for $\mathrm{cl} A$ ?

Chris Trentman

Numerade Educator