Graphics Reference
In-Depth Information
Figure 5.16.
A star-shaped region.
x
0
Section 5.5
5.5.1.
Let
A
be a subspace of a topological space
X
. Prove that
A
is compact if and only if
every cover of
A
by open subsets of
X
has a finite subcover.
5.5.2.
Prove that a closed interval [a,b] in
R
is compact.
Hint:
Consider
A
= {x | [0,x] can be covered by a finite number of sets from the open cover}.
Prove
(1)
0
A
(2)
c = sup
A
A
by finding
a d>0 so that [0,c-d] is compact and [d,c] is contained in an open set from the
cover.)
A
(Find a contradiction to the assumption that 0 < c
(3)
c = 1
Section 5.6
5.6.1.
Prove that a space consisting of two points is not connected.
Prove that every convex subset of
R
n
is connected.
5.6.2.
Use a connectivity argument to justify the fact that a “figure eight” (the wedge
S
1
⁄
S
1
)
5.6.3.
is not homeomorphic to a circle.
Section 5.7
5.7.1.
Show that a space
X
is contractible if and only if the identity map for
X
is homotopic
to a constant map g :
X
Æ
X
.
R
n
. Define
X
to be
star-shaped
if there is some point
x
0
5.7.2.
Let
X
X
, such that for every
x
X
, the segment [
x
0
,
x
] is contained in
X
. See Figure 5.16. Prove that every star-shaped
region is contractible.
5.7.3.
Prove that every cone is contractible. This extends the result from Exercise 5.7.1.
5.7.4.
Prove that a retract of a contractible space is contractible.
