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Title: TopologyQual_Spring2020.
Description: Past paper of TopologyQual_Spring2020.
Description: Past paper of TopologyQual_Spring2020.
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Topology Qualification Exam, Spring 2020
Please attempt 8 out of 9 problems and clearly mark the one you do not want us to grade
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Let f : X ! Y be a surjective, continuous map of topological spaces
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b) Show: if f is a closed map, then it is a quotient map
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Show that a connected metrizable space with at least two points is uncountably infinite
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)
3
...
An isometric embedding ◆ : X ! Y is a
map such that
8x1 , x2 2 X, dY (◆(x1 ), ◆(x2 )) = dX (x1 , x2 )
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a) Show that every isometric embedding from a compact metric space to itself is an
isometry
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)
b) Show that every isometric embedding from Euclidean n-space to itself is an isometry
...
Consider the solid S obtained by digging out the center of a 3-dimensional solid ball and
4 tunnels from the center to the boundary
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5
...
Compute the fundamental group and the homology
groups of X
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Classify the connected 2-fold covering spaces of the Klein bottle K
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)
7
...
8
...
Show that f must have a fixed point
...
Describe the CW structure of X = CP 2 ⇥ RP 2 and use it to compute the homology
groups of X
Title: TopologyQual_Spring2020.
Description: Past paper of TopologyQual_Spring2020.
Description: Past paper of TopologyQual_Spring2020.