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- 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). - Let
*S*^{1}be a circle in**R**^{2}and let*X*be a cylinder*S*^{1}[0, 1] with its subspace topology as a subset of**R**^{3}. Let*Y*be the subset*S*^{1}{0} of*X*and let*Z*be the subset*S*^{1}{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*/*S*^{1}{^{1}/_{2}} ? - 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*S*^{1}with itself is homeomorphic to the sphere*S*^{2}in**R**^{3}. - 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*Y*_{x}=*p*^{-1}({*x*}) for an equivalence class {*x*} in**R**/~ use the fact that any open set containing*Y*_{x}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. - 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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