Download 1 Practice Problems

Math 636
Topology
Paulo Lima-Filho
Assignment # 3
(Due date: Monday, Oct. 11 , 1999)
1
Practice Problems
(You do not need to hand these in.)
Additional Problem 1.
a If T and T 0 are topologies on X, with T finer than T 0 , is it true that a
compact set in T is also compact in T 0 ? What about the converse?
Provide counterexamples whenever necessary.
b Describe the compact sets in the lower limit topology on R. (The reals
with the lower limit topology R` is often called the Sorgenfrey line.)
c Is the Sorgenfrey line connected? Locally connected? Describe its components.
Additional Problem 2.
1. Let X be a (nonempty) compact Hausdorff space. If every point of X
is an acumulation point of X, then X in uncountable.
(HINT:
Step 1 Show that if U is a non-empty subset of X and x ∈ X, there
is a (non-empty) open set V contained in U such that V does not
contain x. Note that x may not be in U.
Step 2 Given f : N → X show that f is not surjective. Start with U =
X a non-empty open set and use previous step to find a non-empty
open set V1 such that V 1 does not contain f (1). Successively define
a nested sequence V 1 ⊃ V 2 ⊃ · · · . Use the compactness of X to
find a point not in the image of f .)
2. Use previous problem to conclude that any closed interval [a, b] in the
real line is uncountable.
Additional Problem 3. A space X satisfies the countable chain condition
iff every disjoint family of open sets is countable. Show that a separable
space satisfies the separable chain condition, but not conversely.
Fall 1999
Assignment # 3
Page 1 of ??
Math 636
Topology
Paulo Lima-Filho
Additional Problem 4. Let {[an , bn ] | n ∈ N} be a collection of intervals
in R such that [an+1 , bn+1 ] ⊂ [an , bn ] for all n. Is it true that ∩n [an , bn ] 6= ∅?
Explain.
Additional Problem 5. Here we construct the Cantor set and study its
basic properties. Let A0 be the closed interval [0, 1] ⊂ R. Let A1 be the
set obtained from A0 by deleting its “middle third” ( 13 , 23 ). Let A2 be the
set obtained from A1 by deleting its “middle thirds” ( 19 , 29 ) and ( 79 , 89 ). In
general, define An by
[ 1 + 3k 2 + 3k .
An = An−1
,
3n
3n
k≥0
The intersection C = ∩n∈N An is the subspace of [0, 1] called the Cantor set.
a A space X is totally disconnected if its only connected subspaces are the
one-point subsets. Show that C is totally disconnected.
b Show that C is compact.
c Show that each An is a union of finitely many disjoint closed intervals of
length 1/3n , and show that the end pointsof these intervals lie in C.
d Show that every point of C is a limit point of C.
e Conclude that C is uncountable.
Fall 1999
Assignment # 3
Page 2 of ??
Math 636
2
Topology
Paulo Lima-Filho
The Problems
You must hand these in.
Problem 1. One says that a Hausdorff space is countably compact if every
countable cover has a finite subcover.
a Give an example of a countably compact space which is not compact.
b Show that X is countably compact if and only if every family of closed
subsets having the finite intersectin property also has the countable
intersection property.
c If Y is first countable and X is countably compact, then a continuous
bijection f : X → Y is a homeomorphism.
Problem 2. Given a subset A ⊂ X of metric space X, define diam(A) =
sup{dist(x, y) | x, y ∈ A}. One says that A is bounded if diam(A) < ∞. Let
A be a compact subspace of X. Show that A is closed and bounded. Give
two non-homeomorphic examples of metric spaces where closed and bounded
does not imply compactness.
Problem 3. Show that a closed subspace of a normal space is normal.
Problem 4. A topological space X is a door space if every subset is either
open of closed. Show that a Hausdorff door space has at most one accumulation point, and if x is a point which is not an accumulation point, then {x}
is open. Give an example of a non-discrete door space.
Problem 5. A point x in a space X is an accumulation point for a subset
A ⊂ X if every neighborhood of x intersects A at a point other than x. In
other words U ∩ (A − {x}) 6= ∅ for every neighborhood U of x.
1. Show that a subset A of X is closed if and only if it contains all its
accumulation points.
2. Prove that x is an accumulation point of A if and only if there is a
net Φ : D → X converging to x and satisfying Φ(α) ∈ A − {x} for all
α ∈ D.
Fall 1999
Assignment # 3
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