TDA in pictures
Detecting topological features with a filtration:
“Barcodes” of persistence homology:
Topological spaces: basic notions
01 Theory - Spaces, subspaces, connectedness
Topological space
A topological space is a set
(the ‘space’) together with a topology structure. A topology structure is a determination of open subsets of
. Formally: open sets are collected together in a set of sets
, the topology. Open sets (in
) must satisfy axioms:
and are open - All unions of opens are open
- Finite intersections of opens are open
A set
is called closed when is open.
Although these axioms for open sets seem rather obscure, a long history (leading up to and in consequence of the axioms) has shown that the concept of open set they determine accurately formalizes our intuitions about spatial connectedness, and the aspect of proximity that underwrites connectedness.
Many geometric sets have natural topology structures:
Some notes:
- The trivial topology has only the minimum of open sets,
and . - The discrete topology has the maximum of open sets,
. (All subsets are open.) - Sometimes a set is simultaneously closed and open, thus ‘clopen’. E.g.
and are always clopen.
Subspace topology
For any subset
, there is a subspace topology structure on . The open sets of
are the intersections where is open in .
Beware that open sets
Connectedness
A topological space
is connected when it is impossible to decompose it as a disjoint union of opens:
02 Illustration
Simple topology - connectedness
Three points:
, with two (nontrivial) open sets. This is not a topology because the intersection is not open:
This is a topology, and it is connected:
Continuum - real number interval
The standard topology structure for real numbers is generated by open intervals
. (“Generated”: it is the smallest topology structure containing all such intervals.)
In
D, the open sets are generated by open boxes or (equivalently) open balls .
Open in
but not in Consider the subspace
equipped with the subspace topology inherited from the topology structure on :
An open set in
may be given as the intersection with a ball in , but this set is not open in because the top and right edges would be included. No open ball can be found around a point in this edge and contained in .
03 Theory - Metric, open balls, continuity
Metric space
A metric space is a set
(the ‘space’) together with a metric structure. A metric structure is a determination of distance between points in the set.
Formally: distance is measured by a function
satisfying: Triangle inequality:
and more basic axioms:
for all
In
Using any distance function, we can define open balls:
Then we can generate a topology on
Metric space topology
Any metric space
has a natural metric space topology structure. The open sets are generated by open balls
. That is, open sets are all unions of open balls:
Every structure of metric space automatically has a natural structure of topological space due to the above definition.
A topology structure formalizes only the notion of connectedness, or one might say “touching vs. separated”, while a metric structure adds a quantitative measure of distance. As such it implies notions like “double the distance” or “
Topological structures that do not come from metric structures can also be useful and interesting. (E.g. in algebraic geometry.) But the notion of “touching” that they express is rather abstracted from the geometrical one we are accustomed to.
“All unions of open balls”
The collection “all unions of open balls” satisfies the axioms of a topology. The first two axioms clearly hold. To check the third axiom, one would have to formalize the concept of ‘generates’, which is applicable to the concept of ‘subbase’. The essential verification is that for any point inside a finite intersection of open balls, there is another open ball around that point which is contained in the intersection. The triangle axiom is needed to confirm this.
LEMMA: open subsets via open balls
In any metric space
, a subset is open if and only if: In words:
is open IFF any in has some open ball around it which is contained in .
This lemma provides a functional characterization of open sets in a metric space.
Proof of LEMMA
Assume the Lemma condition holds; try to prove
is open.
- For each
, write for some open ball over contained in . Then , and is a union of open sets, thus open. Now assume
is open; try to prove the condition holds.
is necessarily the union of some open balls. (Definition of topology structure given a metric.) - Select any
. By 1., this is in some open ball : - Find a ball centered at
which is inside , thus inside .
- Set
- Definition of ball and
implies - Set
- Triangle Inequality: Every
satisfies - Therefore
Continuous functions
A function between topological spaces
is continuous when:
This definition implies that a connected subset cannot be mapped to a disconnected subset by a continuous
For example, the graph of a function with a jump discontinuity:
The disconnected image of
An equivalent definition of continuity can be cast in terms of the metric structure, if the topology comes from a metric:
Continuity and distance
A map
of metric spaces is continuous at when: For all
, there exists such that:
Then
04 Illustration
In Calc I we learn that
Or, in terms of set containment:
Let us compare this to the definition in terms of open sets:
Open-set continuity agrees with
- continuity Assume that
is continuous according to the topological definition.
- Select any
. Let be given. - The interval
is open. - By the topological definition of continuity, the preimage
is open. It contains . - By the Lemma for metric space continuity, find an open ball around
inside . - This ball is an interval
. - All elements of this interval are mapped into
, confirming the - condition. Assume that
is continuous according to the - definition.
- Select any
open set. We must show is open. - Select any
. Define . - Then
, and so for some by metric space continuity. - This
is an interval: - By the
- definition of continuity, produce some so that: - Therefore, whenever
, we know . - So
is open (by definition), is contained in , and contains , confirming the topological continuity condition.
05 Theory - Compactness, homeomorphism, continuity, homotopy
Compactness
A topological space is compact when every open cover has a finite subcover.
The entire real line
The closed interval
is compact.
- Let
be any open cover of . - Define
:
- Let
be the supremum of . - We show
.
- Suppose
and seek a contradiction. - Take some open set
in containing , and an open ball around and contained in . - Move leftwards to
. There is a finite cover of from . - Add
to this cover, and get a finite cover of . - This contradicts the supremum property of
. So we know . - Take some open set
in containing , and an open ball . - There is a finite cover of
because . - Add
to this cover to get a finite cover of . QED
Homeomorphism
A homeomorphism is a map between topological spaces,
, satisfying:
is bijective is continuous is continuous
PROP: ‘Homeomorphism’ simplifies for compact metric spaces
If
is a continuous bijection between metric spaces, and and are both compact, then is necessarily a homeomorphism.
Homotopy
A homotopy is a morphism between continuous maps.
Given
continuous maps, a homotopy is a continuous function: We say
and are homotopic if any homotopy exists between them.
A homotopy is a continuous deformation of
For example, suppose we have two maps
The “winding number” is preserved by
Topological spaces: further notions
06 Theory - Quotients, gluing
Quotient topology
For any partition of
of into cells , , there is a quotient topology structure on the set of cells . The open sets of
are the collections of cells whose unions are open in .
A quotient space is therefore given by ‘collapsing’ some sets of points in
There are many equivalent formulations of this notion, and these two are most common:
- Using an equivalence relation
on , whereby the equivalence classes determine cells of the partition. Then the quotient space is written . - Recall that partitions and equivalence relations are interchangeable notions.
- Using a surjective function
, whereby the preimage fibers determine cells of the partition. Then has the quotient topology when open sets are given by where is open in .
Gluing
Two spaces
and can be glued along specified subsets and using a bijection by defining the quotient topology where identifies with .
07 Illustration
Gluing shapes
- Glue a pair of triangles along an edge.
- Glue a line segment to a triangle, endpoint to vertex.
Real projective space
Consider the unit sphere
. Antipodal points of are those determining a straight line passing through the origin. Define
as the quotient where equivalence classes of are the pairs of antipodal points. Define
similarly by gluing together antipodal points in . We can also describe
using a disk with antipodally glued boundary:
This surface is also derivable by gluing a disk to a Mobius band:
08 Theory - Embedding, limit points, closure, boundary
Embedding
An embedding
is a continuous injective map that induces a homeomorphism .
It is important to emphasize that
For example, a knot in
Exercise - Touching upon
An open interval is mapped into
by such that one end has a limit point on the image of the middle.
The map
is continuous and bijective. Show that is not continuous.
Limit points
Given a subspace
, a point is a limit point of when every open set around touches somewhere besides .
Notes:
- Limit points are sometimes called accumulation points.
- The limit point
might be outside ! - In a metric space, this can be reformulated:
is a limit point of when there is some sequence such that as . This is written . - In other words,
is “approached” from inside .
- In other words,
Interior and exterior
The interior of a set
is the set of points with an open neighborhood contained in . The exterior of a set
is the interior of the complement of .
Closure of a set
The closure
of a set is the union of with all limit points of .
- Equivalently: the closure
is the complement of the union of all open sets in that don’t touch . - Equivalently: the closure
is the intersection of all closed sets containing .
Boundary of a set
The boundary
of a set is the set of points in both and .
- Equivalently: the boundary
is the set of points for which every open neighborhood touches both the interior and the exterior of . - Boundary points are not always in
!
Simplicial complexes
A simplicial complex is an assembly of simplices, which generalize triangles and tetrahedra. Simplices provide “building block” objects in each dimension that may be glued together to form a complex.
Simplices provide a combinatorial method of representing a large class of topological spaces, since the ‘assembly data’ encoding the gluing of simplices into a complex is combinatorial in nature.
Simplicial homology is a homology theory (topological invariant) that is naturally defined in terms of a given simplicial complex. It is easy to find this homology for a space when a representation as simplicial complex is known.
Simplicial homology is isomorphic to other homology theories.
- Singular homology: abstract ‘naturality’; proving invariance is easier
- Simplicial homology: combinatorial; build complexes by gluing faces ‘directly’; stricter gluing rules
- Delta homology: combinatorial, like simplicial; more relaxed gluing rules
- CW homology: combinatorial, like simplicial but gluing cells (balls) not simplices; build CW complexes inductively; very relaxed gluing rules
For smooth manifolds, we also have:
- De Rham cohomology: uses differential forms “closed but not exact”
09 Theory - Simplices and complexes
- A
-simplex is a point (a vertex).
- A
-simplex is a line segment.
- A
-simplex is a triangle (interior included).
- A
-simplex is a tetrahedron.
Geometric simplex
Vertices:
, required to be affinely independent. Geometric simplex:
- Thus
is the “convex hull” of . In other words: all weighted averages of the vertices. - Affinely independent means:
do not lie in any hyperplane (dimension ) of ; equivalently that are linearly independent, or that they generate a vector space of dimension . - A simplex failing this criterion is called degenerate.
- Standard
-simplex : vertices are the unit vectors in .
For example, with
This defines the line segment from
Similarly, with
A simplicial complex can be characterized in several ways:
- Combinatorially
- Geometrically
- Topologically
Simplicial complex - Combinatorial
A combinatorial simplical complex
with a set of ‘vertices’ is a collection of subsets of , called ‘simplices’, satisfying:
- All faces included: If
, then all faces of are in . (And thus faces of faces, all the way down to vertices.) - Touching only at a face: If
, and , then is a face of and a face of . For
, the faces of are the subsets of lacking a single vertex of .
A simplicial complex has a topology. This is given by gluing the disjoint union of constituent simplices along the faces that intersect. (The quotient topology.)
If the simplicial complex is given a geometric representation in
10 Illustration
For example, given a simplex
A simplicial complex
Simplices of dimension
The simplices of top dimension are sometimes called facets (i.e. the biggest ones are called ‘little faces’).
The intersection of two sub-simplices is allowed to be a single face, not two faces (vertices, in this case):
Torus as simplicial complex
It is easy to create a torus by gluing the edges of a square:
But it is not this easy to represent a torus as a simplicial complex. Initial attempts fail:
One can do it with a little more work and complexity:
Every finite simple undirected graph is a simplex.
- No “parallel edges”
- No “one-loops”
In a geometric representation of a simplicial complex, edges should not cross, etc.
Kuratawki: Graph planarity
The graph planarity theorem of Kuratowski says that a graph is planar (can be drawn in
without self intersection) if and only if it does not contain a copy of or :
11 Theory - Subdivision and triangulation
A single topological space (up to homeomorphism) can be represented in many different ways as a simplicial complex:
One can perform “barycentric subdivision” upon a given simplicial complex. This produces a simplicial complex that is homeomorphic to the original one, but the edges are shorter and areas of faces smaller.
A triangulation is a simplicial complex that is homeomorphic to a given smooth manifold. A triangulated manifold is a simplicial complex together with a homeomorphism to a given smooth manifold.
For example, the tetrahedron obtained by gluing four copies of the standard
Triangulations of spheres
For
, all possible triangulation types have been found by Tutte. For
, there is not a known enumeration of triangulation types.
Simplicial homology
Simplicial homology can be defined with “coefficients in
12 Theory - Boundaries, chain complexes
Fix a simplicial complex
Chain groups
For each
, , define the chain group : In other words:
An element of the
-chain group, written generically as , is called a -chain.
- Sometimes
is dropped and we write just when the complex is clear from context.
By allowing combinations of simplices with coefficients, we can define a boundary operator with alternating signs. This alternation leads to cancellation when boundaries of boundaries are taken. The cancellation in turn enables the definition of homology, which is generated by cycles of simplices that are not boundaries.
Boundary operator
The boundary operator
maps simplices to their boundaries. It is defined on simplices and extended linearly to chains:
Examples:
- A
-simplex:
- A
-simplex:
Orientation and vertex ordering
A simplex was initially defined as the convex closure of a set of points. The points were not ordered! The simplex was not ordered!
The boundary map was defined for an ordered list of points (“
omitted”). These definitions are not compatible! Let us upgrade both definitions to employ oriented sets of points.
Hereafter, the notation
will indicate an ordered list of vertex points up to the equivalence relation that identifies lists which are related by an even permutation. For example:
, but: .
- With this convention, the boundary map is well-defined.
- Notice that the orientation of the vertex list determines the sign of its boundary:
- Therefore: one may interpret the sign of a coefficient in
as indicating the orientation of the simplex it is attached to. - Thus the boundary of a 2-simplex (filled triangle) is an oriented path around the triangle:
LEMMA: Boundary of boundary is empty
For all
, we have:
Proof of LEMMA
Apply the double boundary to a generator of
: The second summation duplicates the terms of the first summation, but has one fewer power of
, so the whole sum vanishes. Both summations cover all -simplices omitting some two vertices.
Using this fact we construct the chain complex of group homomorphisms:
Chain complex
Given a simplicial complex
of maximal dimension , we define the chain complex of as the sequence of morphisms: This sequence is a complex because the Lemma ensures that
The image and kernel groups have names:
— called the boundary chains — called the cycles
Homology
The
homology of is defined as the quotient group of cycles modulo boundary chains:
Notice that
13 Illustration
Homology of a triangle
Consider the 1D simplicial complex consisting of a triangle:
The chain complex is:
The group
is generated by the 1-simplices: The group
is generated by the 0-simplices: The boundary operator:
The kernel of
is freely generated by . The image of is isomorphic to via the map determined by . So we have
and . Higher homology groups vanish.
Homology counts holes by dimension
Consider the space
, the wedge sum of three copies of and one copy of obtained by gluing all spaces at a single point:
The homology of this space is:
Homology independent loops in a graph
The “nullity” of a connected graph is the number of independent cycles in the graph. If
for a graph considered as a simplicial complex, then is the nullity. So
for the following graph:
