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Math 598
Jan 21, 2005
1
Geometry and Topology II
Spring 2005, PSU
Lecture Notes 3
1.8
Immersions and Embeddings
Let
X
and
Y
be topological spaces and
f
:
X
→
Y
be a continuous map. We
say that
f
is an
immersion
provided that it is locally onetoone, and
f
is an
embedding
provided that
f
is a homeomorphism between
X
and
f
(
X
) with
respect to the subspace topology.
Exercise 1.
Show that if
X
is compact,
Y
is Hausdorf and
f
:
X
→
Y
is a
onetoone continuous map, then
f
is an embedding. Demonstrate, by means
of a counterexample, that the previous sentence may not be true if
X
is not
compact.
Exercise 2.
Show that if
f
:
M
→
N
is an immersion between manifolds of
the same dimension, then
f
is open. In particular, if
M
is compact, and
N
is connected but not compact, then there exists no immersion
f
:
M
→
N
.
(
Hint:
Use the invariance domain:
if two subsets of
R
n
are homeomorphic
and one of them is open then the other is open as well. )
Exercise 3.
Let
C
⊂
M
be a compact set, and
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 Logic, Geometry, Topology

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