Maths - Topology - Martin Baker

Here we are concerned with shapes, when they are the same shape and the mappings between them.

For example, continuous mappings as allow a toroid to be continuously mapped into a mug.

It is like a sort of 'rubber geometry' where there are no absolute distances.

Determining Same Shape

There are different ways to 'equate' shapes such as homotopy equivalence and homeomorphisms. Homotopy equivalence tells us that a shape can be continuously deformed into another shape. A homeomorphism is a stricter comparison.

For instance a Möbius band is homotopy equivalent to a band but they are not homeomorphic.

Homotopy Equivalence

Homotopy equivalence tells us that a shape can be continuously deformed into another shape. we can see that a Möbius band can be continuously deformed to a non-twisted band by first contracting it continuously to a circle and then continuously expanding it to a band.

diagram

How this can be defined more precisely?

One possibility might be to map every point in X to a point in Y and require an inverse map which brings the point back to where it started.

However this doesn't work when, say, mapping a circle to a line. We can find ways to map every point on the circle to the line and back again but they are different shapes.

We need an extra requirement that every pair of points which are adjacent on X must be adjacent when mapped to Y. In this case there is at least one point where this is not true so this gives a way to show the circle and the line are different shapes.

So we need a way to make this continuity idea more precise.

Continuous Maps

A function is continuous if it doesn't jump, that is, when two inputs of the function get close to each other then the corresponding outputs of the function get close to each other.

Here we discuss the most general form of continuous map which also applies to non-metric topological spaces.

A continuous mapping implies a limited kind of reversibility, at least locally.

Definition 1

Here is one definition of continuity, based on open sets:

Let X and Y be topological spaces. A function f : X->Y is continuous if f^-1(V) is open for every open set V in Y.

Definition 2

Another definition of continuity, based on neighbourhoods, is equivalent to the above definition.

Let X and Y be topological spaces. A function f : X->Y is continuous if for every x

X and every open set U containing f(x), there exists a neighbourhood V of x such that f(V)

More detail about continuity on page here.

Continuous Surjective Maps

Here is an extreme case of a Surjective Map which maps to a single point.

This seems to meet the requirements as there is an open set round the whole preimage. We can think of surjections which meet these requirements as 'nice surjections'.

See fibrations

Continuous Injective Maps

Here is an extreme case of a Injective Map which maps from a single point.

In this case all the open sets in the codomain need to map back to a single open set in the domain. This meets the requirements so it is a 'nice injection'.

See cofibrations

Homotopy

If we have two topological spaces X & Y with two continuous maps between them F & G then a homotopy is a continuous map between F & G.

In the diagram we can draw this in the domain so the functions are homotopic if:

H: X × [0,1] -> Y

where:

H(x,0) = F(x)
H(x,1) = G(x)

We can think of this as 'filling in' the gap between F and G so that we can take any path through it.

Or we can draw it in the codomain (currying) so that each element in X maps to I->Y.

More about homotopy on page here.

Homotopy Equivalence

If we continuously map every point in X to a point in Y and require an inverse map which brings the point back to where it started and do this in both directions we get a homeomorphism.

But if we want a homotopy equivalence (shapes which can be continuously deformed into each other) then we only need a weaker requirement that this round trip and the identity maps are a homotopy.

Homeomorphism

In two dimensions it is relatively easy to determine if two spaces are topologically equivalent. We can check if they:

are connected in the same way.
have the same number of holes.

However, when we scale up to higher dimensions this does not work and it can become impossible to determine homeomorphism. There are methods which will, at least, allow us to prove more formally when topological objects are not homeomorphic.

These methods use 'invariants': properties of topological objects which do not change when going through a homeomorphism. Here we look at two types of invariants which arise from homotopy and homology.

Homotopy

It would take up too much space to properly explain the concept here so see this page for more details

nth homotopy group

If 2 spaces are homotopy equivilant the fundamental groups are isomorphic.

In homotopy we use equivalence classes between a circle and loops (which don't collapse) on the topological object that we are investigating. We can get further invariants by extending the circle to the surface of higher order n-spheres.

We can then get algebraic structures (mostly groups) by investigating what happens when these loops are composed, the loops are generators of the group. In homotopy the order of these compositions can be significant, that is the groups are not necessarily abelian.

Homology

It would take up too much space to properly explain the concept here so see this page for more details

homology groups

In homology we dont just use n-spheres but every closed oriented n-dimensional sub-manifold.

It also uses a different definition of equivalence classes where composition of loops commutes. This results in abelian groups.

So we dont need to fix the basepoint.

Example

Are the two shapes on the right 'homotopy equivalent' ?

Is there a continuous map between them in both directions?

There is more about this example on the page here.

Topology on a Set

A topology on a set X is a collection Τ of subsets of X, called open sets satisfying the following properties:

X and Ø are elements of Τ.
Τ is closed under finite intersections.
Τ is closed under arbitrary unions.

The requirements for the existence of of meets and joins correspond to the requirements for the existence of unions and intersections of open sets. Therefore these lattice structures can represent topologies.

	Lattice	Open Set



invalid - every intersection should be an open set:

Venn Diagram	Topological Space (or not)	Lattice (frame)
	This is not a topological space because 'a' and 'b' are subsets but not the union of 'a' and 'b'
	This is now a topological space because we have added the union of 'a' and 'b'
	This is not a topological space because 'ab' and 'bc' are subsets but not their intersection.
	This is now a topological space because we have added 'b'

Note: Assume Ø is included in the above examples.

Topological Space

In many cases the concept of a metric space is unnecessary, however we still need the concept of 'nearness' and hence 'continuity'. 'Topological space' based on the 'topological open set' is the most general way we can do this. This allows us to define nearness purely using the concept of a subset.

Hausdorff Space

Hausdorff space is a bit more specific than general topological space. Space is Hausdorff if, in addition to being a topological open set, for any two points:
x₁, x₂X there are disjoint open sets U₁, U₂ that contain x₁and x₂.

Basis

A basis is a subcollection B= U

where

U is the open set
B is the basis
= a subcollection (subset) - I need to add the proper symbol for this.

such that evey element of U is a union of open sets in B.

Link between Topology and Logic

One way to visualise the link between topology and logic is to start with a Venn diagram.

We can then map points in the Venn diagram to either true or false depending on whether they are in a given set.

Maths - Algegraic Topology

Determining Same Shape

Homotopy Equivalence

Continuous Maps

Definition 1

Definition 2

Continuous Surjective Maps

Continuous Injective Maps

Homotopy

Homotopy Equivalence

Homeomorphism

Homotopy

Homology

Example

Topology on a Set

Topological Space

Hausdorff Space

Basis

Link between Topology and Logic

Link between Topology and Category Theory

Presheaf

Fibre in Category Theory

Slice Category

Examples

Non-Geometric Examples

Sequences

Denotional Semantics

Recursively Defined Structures

Further Information

EuclideanSpace

Where are you

Related Subjects

Other Subjects

Homology

Inverse Functions