Revision lecture

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Revision lecture

• MA30041 Metric Spaces

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Just to become familiar with the clickerWhat
day of the week is today?

• Sunday
• Monday
• Tuesday
• Wednesday
• Thursday
• Friday
• Saturday

3
Just to become familiar with the clickerThe
concept Metric Space was introduced by whom?

• G. Cantor (1872)
• M.R. Fréchet (1906)
• F. Hausdorff (1914)
• S. Banach (1922)
• I have to phone a friend!

4
Just to become familiar with the clicker Have
you used the Audience Response Systems (ARS)
previously?

• Yes
• No

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The function dist(Y,Z)infd(y,z)y in Y, z in Z
for nonempty subsets Y,Z of a metric space (X,d)
(where dist(Y,Z)8 otherwise), defines a metric.

• True
• False

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A pseudometric differs from a metric by

• (MS1) A pseudometric might not be nonnegative
• (MS2) A pseudometric might not be 0 iff xy
• (MS3) A pseudometric might not be symmetric
• (MS4) Only a modified form weak form of the
  triangle inequality has to hold.

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Let A be a subset of a metric space (X,d). What
is the relationship between diam(int A), diam(A)
and diam(cl A)?

• diam(int A)diam(A)diam(cl A)
• diam(cl A)diam(A)diam(int A)
• diam(int A)diam(A)diam(cl A)

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Let (X,d) be a metric space and (xn) be a
sequence in X. Set Axnn in N. Is the
following statement true or falseIf (xn) is
Cauchy, then the subspace A is totally bounded.

• True
• False

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Let (X,d) be a metric space and (xn) be a
sequence in X. Set Axnn in N. Is the
following statement true or falseIf (xn)
converges to x, then the union of A and x is a
compact set.

• True
• False

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Let A be a subset of a metric space (X,d) s.t.
?AØ. What can you say about A?

• AA
• A has empty interior.
• A is clopen.

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Is the limit of a converging sequence (xn) a
limit point of the set xnn in N?

• Yes, the limit is a limit point.
• Only if the sequence is not eventually constant.
• Yes, if the set is infinite.

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What is the derived set of

• The empty set.
• 0
• . 1/n
• It equals its derived set.

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Let U,V be subsets of a metric space (X,d). If ?U
is a subset of V, and V is a subset of U, then

• ?V is a subset of ?U
• ?U is a subset of ?V
• no general statement about the relationship
  between ?U and ?V is possible

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Quick and dirty solution

• We have ?Ucl U n cl Uc cl U n cl Uc n cl Uc
  ?U n cl Uc.
• But ?U is a subset of V and thus cl V, Uc is a
  subset of Vc, thus cl Uc is a subset of cl Vc
  therefore ?U is a subset of cl V n cl Vc ?V.

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The open subsets of the subspace Rx0 of R2 with
the Euclidean metric are precisely those subsets
Ux0 where U is open in R. Thus, except the
empty set,

• none of these subsets Ux0 is also open in R2
• all of these subsets Ux0 is also open in R2

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??? interval of R is a continuous image of R
itself.What can you replace ??? with?

• Every open
• Every closed
• Every closed and bounded
• Every bounded
• Every

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Some hints

• sin(R) -1,1
• arctan(R)(-p/2,p/2)
• f(x) sin(x) if x0 and f(x)arctan(x) of xgt0 is
  continuous (or f(x)x if )!

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Is a continuous function on a complete metric
space in general bounded?

• Yes, and I know how to prove it.
• No, I have a counterexample.
• No, and I have an argument.
• Not sure.

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Is a continuous bijection between sequentially
compact metric spaces a homeomorphism?

• Depends on the function, however this function is
  automatically uniformly continuous.
• It is false if the function is not uniformly
  continuous.
• Yes, always.

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Which of the following functions is not uniformly
continuous on the specified domain?

• f(x)1/(1x2) on -1,1
• f(x)1/(1x2) on R
• f(x)tan(x) on 0,p/2)
• f(x)x1/x on 1,8)

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Can a complete metric space be a subspace of an
incomplete metric space?

• Yes, but I dont know any example.
• Yes, and I have an example.
• No. However, I forgot the appropriate theorem.
• No, and I know a reason.

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A discrete metric space is sequentially compact
iff

• it is finite and thus totally bounded.
• it is closed and complete (it is always
  bounded!).
• the Cauchy sequences are eventually constant.

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Is every complete metric space sequentially
compact?

• Yes (and totally bounded).
• No, but it is true if we replace sequentially
  compact by compact.
• No, I have a non-(sequentially) compact
  counterexample.

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Let (X,d) be a metric space with subsets U,V.
Is the following statement true or false?If U,
V are compact, then their union is also compact.

• True
• False

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When is a discrete metric space connected?

• Not possible
• If it consists of one point only
• If it consists of two points
• If it consists of finitely many points
• If it consists of infinitely but countably many
  points.

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A ball in a connected metric space need not be
connected.

• True
• False

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Let A be a subset of a metric space (X,d). If the
boundary ?A is connected, is A itself connected?

• Yes
• No

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The checkerboard
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Consider the set
Which statement is false.

• A is path connected and connected.
• A is connected but not path connected.
• cl A is connected, but not path connected.

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Evaluation Using the Audience Response Systems
(ARS) for this revision session was a good idea.

• Strongly Agree
• Agree
• Neutral
• Disagree
• Strongly Disagree