A topology on a set X is a collection τ (tau) 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.
Open and closed sets give us a way to define concepts such as nearness and connectedness on spaces more general than only metric spaces.
Example 1For example this set of open sets on {A,B,C,D} forms a topology: We can verify that it is a valid topology because it contains all the required elements, for example:
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{} |
Graphical Representation as a Space
How can we picture this as a set of points in a space? There may be multiple ways to represent this graphically as there are various dualities such as Poincaré duality and duality between open and closed sets. Here the rectangles represent open sets so we can see which open set each of the elements is in. |
Graphical representation of example 1 |
As a Lattice - Birkhoff's Representation Theorem
| Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets (open sets in this context). A lattice is a partially ordered set in which every pair of elements has a unique supremum (join) and a unique infimum (meet). Here:
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Lattice for example 1 |
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I am trying to work out if we can combine lattices. So here I have embedded example 1 into a full discrete lattice. So we have all meets of {B} and {D} but A and C are injected in at a higher level. |
https://en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem
As a Preorder
Every finite topological space gives rise to a preorder on its points by defining:
x <= y if and only if x belongs to every neighborhood of y.
Every finite preorder can be formed as the specialization preorder of a topological space in this way. That is, there is a one-to-one correspondence between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not one-to-one.
https://en.wikipedia.org/wiki/Preorder
the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest.
https://en.wikipedia.org/wiki/Specialization_(pre)order
Stone Duality
https://en.wikipedia.org/wiki/Stone_duality
Disjoint Union Spaces
https://ncatlab.org/nlab/show/disjoint+union+topological+space
Finite Open Sets
Here we denote the open sets with a finite set of symbols.
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Example 2 - discrete topologyØ = {} |
In this example there are 3 open sets denoted by the blue letters A, B and C. We define these open sets by elements they contain, here denoted by the red letters a, b, c, d, e, f and g. A := {a,d,e,g} |
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Poincaré duality
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On the left of this diagram the blue letters A, B and C represent points, on the right the blue letters represent sets. The lower case red letters a, b, c, d, e, f and g are sets on the left and points on the right. |
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