Download Lecture 9

Lecture 9 - 9/12/2012
Math 5801
General Topology and Knot Theory
Nathan Broaddus
Ohio State University
September 12, 2012
Nathan Broaddus
Lecture 9 - 9/12/2012
General Topology and Knot Theory
Subspace Topology Closed Sets
Course Info
Reading for Friday, September 14
Chapter 2.17, pgs. 92-100
HW 4 for Monday, September 17
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Chapter 2.13: 1, 3, 5, 8a
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Chapter 2.16: 1, 4, 6, 9
Nathan Broaddus
General Topology and Knot Theory
Lecture 9 - 9/12/2012
Subspace Topology Closed Sets
Subspace Topology
Definition 99 (Subspace Topology)
Let X be a topological space and A ⊂ X . The subspace topology on A
is
TA = U ∩ A U open in X .
Proposition 100
Let X be a topological space and A ⊂ X . Then the subspace topology is
a topology on A.
Proof.
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∅ = ∅ ∩ A and A = X ∩ A
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If U, V open in A then there are U 0 , V 0 open in X s.t. U = U 0 ∩ A
and V = V 0 ∩ A. So U ∩ V = (U 0 ∩ A) ∩ (V 0 ∩ A) = (U 0 ∩ V 0 ) ∩ A
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Let U be a collection S
of open sets in X . Then
S
{U ∩ A|U ∈ U} = ( U) ∩ A.
Nathan Broaddus
Lecture 9 - 9/12/2012
General Topology and Knot Theory
Subspace Topology Closed Sets
Subspace Topology
Proposition 101 (Basis for Subspace Topology)
Let X be a topological space with basis B and A ⊂ X . The subspace
topology on A has basis
BA = B ∩ A B ∈ B .
Proposition 102
Let X be a topological and A ⊂ X be an open set in X and U ⊂ A be
open in A. Then U is open in X .
Nathan Broaddus
General Topology and Knot Theory
Lecture 9 - 9/12/2012
Subspace Topology Closed Sets
Closed Sets
Definition 103 (Closed Set)
Let X be a topological space. A subset A ⊂ X is closed if X − A is open.
Examples 104
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In any topological space ∅ and X are closed (and open).
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In the discrete topology every set is closed (and open).
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In the finite complement topology on X finite sets (and X ) are
closed.
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In R (b, ∞) and (−∞, a) are open so [a, b] is closed.
Nathan Broaddus
Lecture 9 - 9/12/2012
General Topology and Knot Theory
Subspace Topology Closed Sets
Closed Sets
Definition 105 (Interior and Closure)
Let X be a topological space and A ⊂ X .
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The interior of A is the union of all open subsets of A.
[ Int A = A◦ =
U U open in X and U ⊂ A .
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The closure of a A is the intersection of all closed sets containing A.
\ Cl A = A =
C C closed in X and A ⊂ C .
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Int A exists by (UNION) axiom of set theory.
Int A is open since it is a union of open sets.
Int A is the largest open subset of A
A exists since X is a closed set containing A and (SEP) axiom.
A is closed since it is an intersection of closed sets.
A is the smallest closed set containing A.
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Nathan Broaddus
General Topology and Knot Theory
Lecture 9 - 9/12/2012
Subspace Topology Closed Sets
Closed Sets
Examples 106 (Closures and Interiors)
1. In the discrete topology Xd we have A = Int A = A.
2. In R we have Int[1, 2) = (1, 2) and [1, 2) = [1, 2].
3. What is the closure of (1, 2) in Rf ? R
4. What is the interior of (1, 2) in Rf ? ∅
5. What are the closure and interior of {3, 5, 12} in Zf ?
{3, 5, 12} and ∅
6. What are the closure and interior of {5n|n ∈ Z} in Zf ?
Z and ∅
7. What are the closure and interior of Z in R?
Z and ∅
8. What are the closure and interior of Q in R?
R and ∅
Nathan Broaddus
Lecture 9 - 9/12/2012
General Topology and Knot Theory
Subspace Topology Closed Sets
Closed Sets
Definition 107 (Denseness)
Let X be a topological space and A ⊂ X . A is dense in X if A = X .
Examples 108 (Denseness)
1. Is Q is dense in R? Yes.
2. Is Z dense in R? No.
3. Is Z dense in Rf ? Yes.
Definition 109 (Limit Point)
Let X be a topological space and A ⊂ X . x ∈ X is a limit point of A if
x ∈ A − {x}.
Nathan Broaddus
General Topology and Knot Theory
Lecture 9 - 9/12/2012
Subspace Topology Closed Sets
Closed Sets
Examples 110 (Limit Points)
1. In R 3 is a limit point of (1, 3).
2. In R 3 is a limit point of (1, 3].
3. In Xd ∀x ∈ X and ∀A ⊂ X x is not a limit point of A.
4. In R is 3 a limit point of Z? No.
5. In Rf is π is a limit point of Z? Yes.
6. In R is π a limit point of Q? Yes.
7. In R is
1
8
a limit point of A = { n1 |n ∈ Z+ }? No.
8. In R is 0 a limit point of A = { n1 |n ∈ Z+ }? Yes.
Nathan Broaddus
Lecture 9 - 9/12/2012
General Topology and Knot Theory
Subspace Topology Closed Sets
Closed Sets
Definition 111 (Boundary)
If X is a space and A ⊂ X then the boundary or frontier of A is
Bd A = Fr A = ∂A = A ∩ X − A
Examples 112 (Boundaries)
1. In R Bd(1, 3] = {1, 3}.
2. In Xd ∀A ⊂ X Bd A = ∅.
3. In Xt ∀A ⊂ X If A 6= ∅ and X − A 6= ∅ then Bd A = X .
4. In R what is Bd Z? Z
5. In Rf what is Bd Z? R
6. In R what is Bd Q? R
7. In R what is Bd{ n1 |n ∈ Z+ }? {0} ∪ { n1 |n ∈ Z+ }
Nathan Broaddus
General Topology and Knot Theory
Lecture 9 - 9/12/2012
Subspace Topology Closed Sets
Closed Sets
Definition 113 (Neighborhood)
If X is a space a neighborhood of x is a set A such that there is an
open set U with x ∈ U ⊂ A.
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Note book def. insists that neighborhoods be open sets.
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book “neighborhood” = lecture “open neighborhood”
Examples 114 (Neighborhoods)
1. In R Bd(1, 3] is not a neighborhood of 3.
2. In R [2, π] is a neighborhood of 3.
3. In X ∀x ∈ X X is a nbhd. of x.
Nathan Broaddus
General Topology and Knot Theory