On these pages we discuss some concepts in algegraic topology. Click on the subject on the right to go to the subject you are interested in. |
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Homotopy Equivalence
In two dimensions it is relatively easy to determine if two spaces are topologically equivalent (homeomorphic). 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.
HomotopyIt would take up too much space to properly explain the concept here so see this page for more details |
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. |
HomologyIt would take up too much space to properly explain the concept here so see this page for more details |
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. |
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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 | |
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| invalid - every intersection should be an open set: |  |
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| Venn Diagram | Topological Space (or not) | Lattice (frame) |
|---|---|---|
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This is not a topological space because 'a' and 'b' are subsets but not the union of 'a' and 'b' |  |
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This is now a topological space because we have added the union of 'a' and 'b' |  |
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This is not a topological space because 'ab' and 'bc' are subsets but not their intersection. |  |
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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, x2
X 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. |
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This is a nice way to link geometry, logic and topology.
see pages about:
Link between Topology and Category Theory
Presheaf
Fibre in Category Theory
See page here.
Slice Category
See page here.
Examples
Non-Geometric Examples
Sequences
Here the concept of 'nearness' comes from sequences that begin in the same way.
Denotional Semantics
Recursively Defined Structures
Further Information
see pages about:
