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Topology: An Inquiry-Based Approach

Steven Schlicker

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Exercises Exercises

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1.

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Determine, with proof, which of the following sets A is a neighborhood of a in the indicated metric space.
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(a)

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A={(x,y)∈R2∣x2+y2<1} in (R2,dE) with a=(0.5,0.5)
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(b)

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A is the x-axis in (R2,dT) with a=(0,0), where dT is the taxicab metric
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(d)

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A is the set of positive integers in (Q,d) and a=1, where Q is the set of all rational numbers in reduced form with metric d:Q×Q→R defined by
d(ab,cd)=max{|aβˆ’c|,|bβˆ’d|}
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(The fact that d is a metric is the topic of Exercise 3.)
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2.

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Let X={1,3,5} and define dX:XΓ—Xβ†’R by dX(x,y)=xyβˆ’1(mod8). That is, dX(x,y) is the remainder when xyβˆ’1 is divided by 8. That dX is a metric on X is examined in Exercise 2. Let (Y,dY) be a metric space. Is it possible to define a function f:Xβ†’Y that is not continuous? Explain.
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3.

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If x=(x1,x2,…,xn), we let |x|=x12+x22+β‹―+xn2. For x=(x1,x2,…,xn) and y=(y1,y2,…yn), define dH:RnΓ—Rnβ†’R by
dH(x,y)={0 if x=y|x|+|y| otherwise .
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The fact that dH is a metric is examined in Exercise 7. Let (X,dX)=(R2,dH) and let (Y,dY)=(R,dE). Define f:Xβ†’Y and g:Xβ†’Y by
f(x)={0 if x=(0,0)1 otherwise   and  g(x)={0 if |x|<11 otherwise .
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One of f, g is continuous and the other is not. Determine which is which, with proof for each.
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4.

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Recall from Chapter 3 that we can construct a finite metric space by starting with a finite set of points and making a graph with the points as vertices. We construct edges so that the graph is connected (that is, there is a path from any one vertex to any other) and give weights to the edges. We then define a metric d on S by letting d(x,y) be the length of a shortest path between vertices x and y in the graph. Consider the metric space (X,d) corresponding to the graph in Figure 7.7.
Figure 7.7. A graph to define a metric.
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(a)

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Determine all of the open balls B(a,Ξ΄) for every positive real number Ξ΄.
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5.

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(a)

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Let f:(R,dE)β†’(R,dE) be defined by f(x)=ax+b for some real numbers a and b with aβ‰ 0. Let p∈R and let r>0. Show that fβˆ’1(B(f(p),r)) contains an open ball centered at p. Conclude that every linear function from (R,dE) to (R,dE) is continuous.
Hint.
By Exercise 5 we can assume a>0 to simplify the problem.
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(b)

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Let f:(R,dE)β†’(R,dE) be defined by f(x)=ax2+bx+c for some real numbers a, b, and c with aβ‰ 0. Let p∈R and let r>0. fβˆ’1(B(f(p),r)) contains an open ball centered at p. Conclude that every quadratic function from (R,dE) to (R,dE) is continuous.
Hint.
Consider cases.
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6.

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Let (X,d) be a metric space, and let A be a nonempty subset of X. Exercise 11 tells us that
d(b,A)≀d(b,c)+d(c,A)
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for all b,c∈X. Define f:Xβ†’R by f(x)=d(x,A). Let b∈X. Given Ο΅>0, show that there is a neighborhood N of b such that x∈N implies f(x)∈B(f(b),Ο΅). Conclude that f is a continuous function. (Assume the metric on R is the Euclidean metric.)
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7.

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Let a and b be distinct points of a metric space X. Prove that there are neighborhoods Na and Nb of a and b respectively such that Na∩Nb=βˆ….
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8.

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Let (X,d) be a metric space and let a∈X. Prove each of the following.
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(b)

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If N is a neighborhood of a and NβŠ†M, then M is a neighborhood of a.
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9.

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Let f:(R,dE)β†’(R,dE) be a continuous function. Show that if f(a)>0 for some a∈R, then there is a neighborhood N of a such that f(x)>0 for all x∈N.
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10.

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Let (X,d) be a metric space where d is the discrete metric. Show that every subset of X is a neighborhood of each of its points.
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11.

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For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
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(a)

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If N is a neighborhood of a point a in a metric space X, then any open ball contained in N is also a neighborhood of a.
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(b)

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If N is a neighborhood of a point a in a metric space X, then N is a neighborhood of each of its points.
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(c)

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If X and Y are metric spaces and f:X→Y is a continuous function, then f(N) is a neighborhood of f(a) in Y whenever N is a neighborhood of a in X.
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(d)

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If X and Y are metric spaces and f:Xβ†’Y is continuous at a∈X, and N is a neighborhood of f(a) in Y, then fβˆ’1(N) is a neighborhood of a in X.
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(e)

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If a is a point in a metric space X and if Ξ΄ is a positive real number, then the open ball B(a,Ξ΄) contains infinitely many points of X.
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(f)

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If N1, N2, …, Nk are neighborhoods of a point a in a metric space X for some positive integer k, then β‹‚i=1kNi is a neighborhood of a.
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(g)

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If NΞ± is a neighborhood of a point a in a metric space X for all Ξ± in some indexing set I, then β‹‚Ξ±βˆˆINΞ± is a neighborhood of a.
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