Metric and Topological Spaces

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## Exercises 7

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

2. Let S1 be a circle in R2 and let X be a cylinder S1 [0, 1] with its subspace topology as a subset of R3. Let Y be the subset S1 {0} of X and let Z be the subset S1 {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 {1/2} ?

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 Y, is the space obtained by identifying the points x and y in X Y. i.e. the space (X Y)/{x, y}. Show that this space is homeomorphic to the subspace {x} Y X {y} of X Y.
The smash product of spaces X and Y, written X Y, is the space X Y / X Y.
Prove that the smash product of a circle S1 with itself is homeomorphic to the sphere S2 in R3.

4. Define an equivalence relation on R with the usual topology by x ~ y if and only if x - y Q.
Let p : R 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 .
Hence show that R/~ with the identification topology is homeomorphic to R with the trivial topology.

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 ?

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JOC February 2004