§ Finite topologies and DFS numbering
In this great math overflow question on
How to think about non hausdorff topologies in the finite case , there's an answer that encourages us
to think of them as preorders, which are basically graphs. I wanted to understand
this perspective, as well as connect it to DFS numbers, since they provide a
nice way to embed these topologies into .
§ Closure axioms of topology
We can axiomatize a topology using the kurotawski closure axioms. We need an
idempotent monotonic function which satisfies some
technical conditions. Formally:
Under this, a set is closed if it is a fixed point of : That is, a set
is closed iff .
- [ is a strict function: it takes bottoms to bottoms ]
- . [monotonicity ]
- is idempotent: . [idempotence ]
- for all in , .
§ Slight weakening into Single axiom version
Interestingly, this also gives a single axiom version of topological axioms,
something that maybe useful for machine learning. The single axiom is that
for all , .
This does not provide that , but it does provide
the other axioms [2-4 ].
§ Continuous functions
A function is continuous iff for every .
TODO: give examples of why this works, and why we need and not just .
§ Finite topologies as preorders
We draw an arrow iff . Alternatively stated,
draw an arrow iff . That is, we have an injection
from the closure of into the closure of , and the arrow represents
the injection. Alternatively, we can think of
this as ordering the elements by "information". A point has less information
than point if its closure has fewer points.
§ T0 in terms of closure:
- is iff for points , we have an open set which contains one of the points but not the other. Formally, either , or .
- Example of a space is the sierpinski space . Here, we have the open set by considering the computation
f(thunk) = force(thunk). For more on this perspective, see [Topology is really about computation ](#topology-is-really-about-computation--part-1). This open set contains only and not .
- Closure definition: is iff
§ T1 in terms of closure:
- is iff for all in , we have open sets such that are open neighbourhoods of which do not contain the "other point". Formally, we need and , and similarly and . That is, and can split apart, but and need not be disjoint.
- Example of is the zariski/co-finite topology on , where the open sets are complements of finite sets. Given two integers , use the open sets as the complements of the closed finite sets , and . These separate and , but have huge intersection: .
- Closure definition: is iff .
- T1 has all singleton sets as closed
§ Haussdorf (T2) in terms of closure
- ix iff for all in , we have open set such that they are disjoint ( ) and are neighbourhoods of , : and .
- Example of a space is that of the real line, where any two points can be separated with epsilon balls with centers and radii .
- Closure definition : is iff implies there is a set such that where is the set complement.
§ Relationship between DFS and closure when the topology is
If the topology is , then we know that the relation will be a poset,
and hence the graph will be a DAG. Thus, whenever we have , we
§ DFS: the T0 case
§ DFS: the back edges