Metric and Topological Spaces

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(Exercises 8)

Exercises 7

  1. Make a twisted cylinder by gluing the (short) sides of a strip after a full (360 degrees) twist. Prove that the resulting space is homeomorphic to the cylinder (made by gluing the strip with no twist).

    Solution to question 1

  2. Let S1 be a circle in R2 and let X be a cylinder S1 cross [0, 1] with its subspace topology as a subset of R3. Let Y be the subset S1 cross {0} of X and let Z be the subset S1 cross {0, 1} of X.
    Describe the sets X / Y and X / Z with their identification topologies and identify them as more familiar topological spaces.
    What is the space X / S1 cross {1/2} ?

    Solution to question 2

  3. Let X and Y be disjoint topological spaces with x a point in X and y a point in Y. Then the one-point union or wedge of X and Y, written X wedge Y, is the space obtained by identifying the points x and y in X union Y. i.e. the space (X union Y)/{x, y}. Show that this space is homeomorphic to the subspace {x} cross Y union X cross {y} of X cross Y.
    The smash product of spaces X and Y, written X smash Y, is the space X cross Y / X wedge Y.
    Prove that the smash product of a circle S1 with itself is homeomorphic to the sphere S2 in R3.

    Solution to question 3

  4. Define an equivalence relation on R with the usual topology by x ~ y if and only if x - y belongs Q.
    Let p : R rarrow R/~ be the natural map. Show that the image of any open interval in R is the whole of R/~.
    If Yx = p-1({x}) for an equivalence class {x} in R/~ use the fact that any open set containing Yx must contain an interval to deduce that the only open sets in R/~ with the identification topology are R/~ and empty.
    Hence show that R/~ with the identification topology is homeomorphic to R with the trivial topology.

    Solution to question 4

  5. Make two surfaces A and B by joining two sheets of paper by twisted strips as shown below.

    What can you say about the spaces you get ?
    What would happen if we reversed the twist on one of the strips or if one of the strips was untwisted ?

    Solution to question 5

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(Exercises 8)

JOC February 2004