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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 if the following sets S are open in the subspace A of the topological space (R,dE).
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2.

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Let O be an open set in a metric space (X,d). Show that a subset U of O is open in (O,d|O) if and only if U is open in (X,d).
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3.

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Let (X,dX) and (Y,dY) be metric spaces, and let f:X→Y be a continuous function. If A is a subspace of X, must the restriction f|A of f to A mapping A to Y be continuous? Give a proof that the restriction is continuous, or an example to show that the restriction need not be continuous.
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4.

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Prove Theorem 11.3. That is, let (X,d) be a metric space and A a subset of X. Prove that a subset CA of A is closed in A if and only if there is a closed set CX in X so that CA=CX∩A.
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5.

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Let (X,dX) and (Y,dY) be metric spaces. Prove or disprove: the function d:X×Y→R defined by
d((x1,y1),(x2,y2))=dX(x1,x2)+dY(y1,y2)
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is a metric on X×Y.
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6.

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Let (Xi,di) be metric spaces for i from 1 to some positive integer n. Let d:∏i=1nXi→R be defined
d(x,y)=∑i=1ndi(xi,yi)2
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when x=(x1,x2,…,xn) and y=(y1,y2,…,yn) are in X. Show that d is a metric on ∏i=1nXi.
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7.

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Let (xn) be a non-decreasing sequence of real numbers that is bounded above. That is, xn≤xn+1 for every n and there is a positive real number K such that xn≤K for every n. Show that the sequence (xn) converges.
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8.

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It is possible to consider infinite products as metric spaces. One important example is a Hilbert space H, which consists of all infinite sequences (xn) where xn∈R for every n and ∑k=1∞xk2 is finite. Hilbert space has important applications in physics, particularly in quantum mechanics.
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(a)

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Give two distinct elements in H and one infinite sequence that is not in H. Explain your examples.
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(b)

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We define the norm of an element x=(xn) in H as
‖x‖=∑k=1∞xk2.
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From this norm we can define a distance between elements x=(xn) and y=(yn) in H as follows:
d(x,y)=‖x−y‖,
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where x−y=(xn−yn). Another way to write d is
d(x,y)=∑k=1∞(xk−yk)2.
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One potential problem with this function d is that we need to know that if x and y are in H, then x−y∈H. That is, show that if ∑k=1∞xk2 and ∑k=1∞yk2 are finite, then ∑k=1∞(xk−yk)2 is also finite.
Hint.
Consider a finite sum and use Exercise 7.
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(c)

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Show that d is a metric on H.
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(d)

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Let Em={(xn)∈H∣xk=0 for k>m}. Let f:Em→Rm be defined by f((xn)n=1∞)=(xn)n=1m. Show that f is a bijection such that d((xn),(yn))=dE(f((xn)),f((yn))) for any elements (xn), (yn) in H. So Em is essentially the same as Rm and so we can consider the space Rm as embedded in H as a subspace of H for every m∈Z+.
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9.

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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 d is the discrete metric on a metric space X, then for any subspace A of X, the restriction of d to A is the discrete metric.
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(b)

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If d is a metric on a space X that is not the discrete metric, and if A is a subset of X, then d|A cannot be the discrete metric.
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(c)

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Let A be a subspace of a metric space (X,d). If a sequence (an) is in A and liman=a for some a∈A, then liman=a in X.
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(d)

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Let A be a subspace of a metric space (X,d). If a sequence (an) is in A and liman=a for some a∈X, then liman=a in A.
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(e)

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If (X,dX) and (Y,dY) are metric spaces, then the function d:X×Y→R defined by
d((x1,y1),(x2,y2))=max{dX(x1,x2),dY(y1,y2)}
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is a metric on X×Y.
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(f)

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If (X,dX) and (Y,dY) are metric spaces, then the function d:X×Y→R defined by
d((x1,y1),(x2,y2))=dX(x1,x2)dY(y1,y2)
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is a metric on X×Y.
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