Algebraic Topology
Lecture 1 - 04-09-2023
Quiz
Question 1 A topological space is a set X together with a topology, which is a family T of so-called
open subsets X that satisfy the following properties:
1. ∅, X ∈ T
2. ∀ U, V ∈ T , U ∩ V ∈ T
3. ∀ U, V ∈ T , U ∪i∈I V ∈ T
A subset Z of X is called closed if X\Z is open.
Question 2 Let X be a topological space with topology T and Z a subset of X. The subspace topology
for Z is the topology
TZ = {U ∩ Z | U ∈ T }
Question 3 Let X be a topological space with topology T . A subset S of T generates T (equivalently,
S is called a subbasis) if every U ∈ T is a finite intersection of unions of open subsets of S.
A subset S of T is called a basis of T if every U ∈ T is a union of open subsets of S.
Question 4 A basis for the topology on R is given by
S = {(a, b) | a, b ∈ R}
Question 5 Let X be a metric space with distance function d : X × X → R≥0 . A basis for the topology
of X is given by all open subsets of the form Bϵ (x) = {y ∈ X | d(x, y) < ϵ}.
Question 6 Let X be a topological space, Z a subset of X and x ∈ X.
1. The open interior of Z is
Z ◦ = {x ∈ Z | ∃U ∈ T with x ∈ U ⊂ Z}
2. The closure of Z is
Z = {x ∈ X | ∀U ∈ T with x ∈ U, U ∩ Z ̸= ∅}
3. The boundary of Z is
∂Z = Z\Z ◦
4. Z is a neighbourhood of x if x ∈ Z ◦
5. x is a limit point of Z if for all neighbourhoods U of x, (U ∩ Z)\{x} =
̸ ∅
6. Z is dense in X if Z = X
Question 7 Let X and Y be topological spaces. A continuous function from X to Y is a map f : X → Y
such that for all U ∈ Y open, f −1 (U ) ∈ X is also open.
A homeomorphism from X to Y is a continuous function f : X → Y such that f is bijective and f −1 is
a continuous function.
X and Y are called homeomorphic if a homeomorphism f : X → Y exists.
Question 8 Let X be a topological space. An open cover of X is a family {Ui }i∈I of open subsets of X
such that each Ui is open and X is contained in the infinite union.
1
,Question 9 Let X be a topological space. X is compact if every open cover has a finite subcover. A
compact subset of X is a subset Z such that Z is compact with respect to the subspace topology. The
theorem of Heine-Borel states that a subset Z of Rn is compact if and only if it is closed and bounded.
Question 10 A topological space X is Hausdorff if for all x, y ∈ X, there exists open U, V ∈ T such
that x ∈ U and y ∈ V , but U ∩ V = ∅
Question 11 The theorem of Bolzano-Weierstrass states that every infinite subset of a compact topo-
logical space has a limit point.
Question 12 Lebesque’s lemma states that for every compact metric space X and every open cover
{Ui } of X there is a δ > 0 such that for all x ∈ X, Bδ (x) ⊂ Ui for some i
Q
Question 13 Let {Xi }i∈I be topological spaces. Their product
Q is the Cartesian product i∈I Xi together
with the topology generated by open subsets of the form i∈I Ui where Ui ⊂ Xi are open and Ui = Xi
for all but finitely many i ∈ I
Question 14 Let {Xi }i∈I be a family of compact topological spaces. Tychonoff’s theorem states that
the product of compact topological spaces is compact
Question 15 A topological space X is connected if it admits no partition (U1 , U2 open and neither the
empty set).
A connected component of X is connected with respect to the subspace topology and is maximal.
A path in X is a continuous map p : [0, 1] → X
X is path-connected if for all x, y ∈ X, there exists a path γ such that γ(0) = x and γ(1) = y
If X is connected, then it is path-connected. However, if X is path-connected, then X is not necisarrily
connected.
Lecture notes lecture 1
Some topological objects:
• Sphere
• Torus
• Half-sphere = sphere with 1 disc removed
• Cylinder = sphere with 2 disks removed
• Möbius band
• Twice twisted band
What will we do in this course?
1. What does it mean to deform a space? topological space, continuous maps, homeomorphism.
2. tools to distinct different surfaces? invariants. 3 ways:
(a) elementary properties (looking at the boundary, connectivity properties, etc)
(b) homotopy/fundamental groups (loops up to moving)
(c) homology (invariants of triangulations)
2
, Lecture 2 - 06-09-2023
Chapter 4
4.1 Constructing a Möbius strip
To construct a Möbius strip, one takes a rectangle and identifies a pair of opposite edges with a half
twist. Mathematically: For the rectangle, take the subspace R of E2 consisting of those points (x, y) for
which 0 ≤ x ≤ 3 and 0 ≤ y ≤ 1. To describe the identification of the vertical edges of R with a half
twist, we partition R into disjoint nonempty subsets in such a way that two points lie in the same subset
if and only if we wish them to be identified. The appropriate partition of R consists of:
• sets consisting of a pair of points of the form (0, y), (3, 1 − y) where 0 ≤ y ≤ 1
• sets consisting of a signle point (x, y) where 0 < x < 3 and 0 ≤ y ≤ 1
We have now defined a set M , its points being the subsets of the partition. π : R → M is the natural
function that sends each point of R to the partition in which it lies. The identification topology on M
is defined to be the largest topology for which π is continuous. Next we label with L the image under π
of the two vertical edges of R. Let R∗ = R\π −1 (L), then the restriction of π to R∗ is a homeomorphism
of R∗ with M \L. If p lies on L then π −1 (p) consists of two disctict points on the vertical edges of R of
the form (0, y), (3, 1 − y). The union of two open half-discs in R with centers (0, y), (3, 1 − y) maps to an
open neighourhood of p in the identification topology.
Lecture 3 - 11-09-2023
Chapter 4
4.2 The identification topology
S
Let X be a topological space and let P be a family of disjoint nonempty subsets of X such that P = X.
We then call P a partition of X. We can form an identification space Y as follows:
1. The points of Y are the members of P
2. π : X → Y sends each point of X to the subset of P containing it
3. The topology of Y is the largest for which π is continuous. We call this topology the identification
topology.
Theorem 4.1 Let Y be an identification space and let Z be an arbitrary topological space. A function
f : Y → Z is continuous if and only if the composition f π : X → Z is continuous
3
Lecture 1 - 04-09-2023
Quiz
Question 1 A topological space is a set X together with a topology, which is a family T of so-called
open subsets X that satisfy the following properties:
1. ∅, X ∈ T
2. ∀ U, V ∈ T , U ∩ V ∈ T
3. ∀ U, V ∈ T , U ∪i∈I V ∈ T
A subset Z of X is called closed if X\Z is open.
Question 2 Let X be a topological space with topology T and Z a subset of X. The subspace topology
for Z is the topology
TZ = {U ∩ Z | U ∈ T }
Question 3 Let X be a topological space with topology T . A subset S of T generates T (equivalently,
S is called a subbasis) if every U ∈ T is a finite intersection of unions of open subsets of S.
A subset S of T is called a basis of T if every U ∈ T is a union of open subsets of S.
Question 4 A basis for the topology on R is given by
S = {(a, b) | a, b ∈ R}
Question 5 Let X be a metric space with distance function d : X × X → R≥0 . A basis for the topology
of X is given by all open subsets of the form Bϵ (x) = {y ∈ X | d(x, y) < ϵ}.
Question 6 Let X be a topological space, Z a subset of X and x ∈ X.
1. The open interior of Z is
Z ◦ = {x ∈ Z | ∃U ∈ T with x ∈ U ⊂ Z}
2. The closure of Z is
Z = {x ∈ X | ∀U ∈ T with x ∈ U, U ∩ Z ̸= ∅}
3. The boundary of Z is
∂Z = Z\Z ◦
4. Z is a neighbourhood of x if x ∈ Z ◦
5. x is a limit point of Z if for all neighbourhoods U of x, (U ∩ Z)\{x} =
̸ ∅
6. Z is dense in X if Z = X
Question 7 Let X and Y be topological spaces. A continuous function from X to Y is a map f : X → Y
such that for all U ∈ Y open, f −1 (U ) ∈ X is also open.
A homeomorphism from X to Y is a continuous function f : X → Y such that f is bijective and f −1 is
a continuous function.
X and Y are called homeomorphic if a homeomorphism f : X → Y exists.
Question 8 Let X be a topological space. An open cover of X is a family {Ui }i∈I of open subsets of X
such that each Ui is open and X is contained in the infinite union.
1
,Question 9 Let X be a topological space. X is compact if every open cover has a finite subcover. A
compact subset of X is a subset Z such that Z is compact with respect to the subspace topology. The
theorem of Heine-Borel states that a subset Z of Rn is compact if and only if it is closed and bounded.
Question 10 A topological space X is Hausdorff if for all x, y ∈ X, there exists open U, V ∈ T such
that x ∈ U and y ∈ V , but U ∩ V = ∅
Question 11 The theorem of Bolzano-Weierstrass states that every infinite subset of a compact topo-
logical space has a limit point.
Question 12 Lebesque’s lemma states that for every compact metric space X and every open cover
{Ui } of X there is a δ > 0 such that for all x ∈ X, Bδ (x) ⊂ Ui for some i
Q
Question 13 Let {Xi }i∈I be topological spaces. Their product
Q is the Cartesian product i∈I Xi together
with the topology generated by open subsets of the form i∈I Ui where Ui ⊂ Xi are open and Ui = Xi
for all but finitely many i ∈ I
Question 14 Let {Xi }i∈I be a family of compact topological spaces. Tychonoff’s theorem states that
the product of compact topological spaces is compact
Question 15 A topological space X is connected if it admits no partition (U1 , U2 open and neither the
empty set).
A connected component of X is connected with respect to the subspace topology and is maximal.
A path in X is a continuous map p : [0, 1] → X
X is path-connected if for all x, y ∈ X, there exists a path γ such that γ(0) = x and γ(1) = y
If X is connected, then it is path-connected. However, if X is path-connected, then X is not necisarrily
connected.
Lecture notes lecture 1
Some topological objects:
• Sphere
• Torus
• Half-sphere = sphere with 1 disc removed
• Cylinder = sphere with 2 disks removed
• Möbius band
• Twice twisted band
What will we do in this course?
1. What does it mean to deform a space? topological space, continuous maps, homeomorphism.
2. tools to distinct different surfaces? invariants. 3 ways:
(a) elementary properties (looking at the boundary, connectivity properties, etc)
(b) homotopy/fundamental groups (loops up to moving)
(c) homology (invariants of triangulations)
2
, Lecture 2 - 06-09-2023
Chapter 4
4.1 Constructing a Möbius strip
To construct a Möbius strip, one takes a rectangle and identifies a pair of opposite edges with a half
twist. Mathematically: For the rectangle, take the subspace R of E2 consisting of those points (x, y) for
which 0 ≤ x ≤ 3 and 0 ≤ y ≤ 1. To describe the identification of the vertical edges of R with a half
twist, we partition R into disjoint nonempty subsets in such a way that two points lie in the same subset
if and only if we wish them to be identified. The appropriate partition of R consists of:
• sets consisting of a pair of points of the form (0, y), (3, 1 − y) where 0 ≤ y ≤ 1
• sets consisting of a signle point (x, y) where 0 < x < 3 and 0 ≤ y ≤ 1
We have now defined a set M , its points being the subsets of the partition. π : R → M is the natural
function that sends each point of R to the partition in which it lies. The identification topology on M
is defined to be the largest topology for which π is continuous. Next we label with L the image under π
of the two vertical edges of R. Let R∗ = R\π −1 (L), then the restriction of π to R∗ is a homeomorphism
of R∗ with M \L. If p lies on L then π −1 (p) consists of two disctict points on the vertical edges of R of
the form (0, y), (3, 1 − y). The union of two open half-discs in R with centers (0, y), (3, 1 − y) maps to an
open neighourhood of p in the identification topology.
Lecture 3 - 11-09-2023
Chapter 4
4.2 The identification topology
S
Let X be a topological space and let P be a family of disjoint nonempty subsets of X such that P = X.
We then call P a partition of X. We can form an identification space Y as follows:
1. The points of Y are the members of P
2. π : X → Y sends each point of X to the subset of P containing it
3. The topology of Y is the largest for which π is continuous. We call this topology the identification
topology.
Theorem 4.1 Let Y be an identification space and let Z be an arbitrary topological space. A function
f : Y → Z is continuous if and only if the composition f π : X → Z is continuous
3
