A Universe of Sorts

§ 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

\mathbb R

§ Closure axioms of topology

We can axiomatize a topology using the kurotawski closure axioms. We need an idempotent monotonic function

c: 2^X \rightarrow 2^X

which satisfies some technical conditions. Formally:

$c(\emptyset) = \emptyset)$ [ $c$ is a strict function: it takes bottoms to bottoms ]
$A \subseteq c(A)$ . [monotonicity ]
$c$ is idempotent: $c(c(A)) = c(A)$ . [idempotence ]
for all $A, B$ in $X$ , $c(A \cup B) = c(A) \cup c(B)$ .

Under this, a set is closed if it is a fixed point of

c

: That is, a set

A

is closed iff

c(A) = A

§ 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

A, B \subseteq X

A \cup c(A) \cup c(C(B)) \subseteq c(A \cup B)

. This does not provide that

c(\emptyset) = \emptyset

, but it does provide the other axioms [2-4 ].

§ Continuous functions

A function is continuous iff

f(c(A)) \subseteq c'(f(A))

for every

A \in X

. TODO: give examples of why this works, and why we need

(\subseteq)

and not just

(eq)

§ Finite topologies as preorders

We draw an arrow

x \rightarrow y

iff

x \in Closure(y)

. Alternatively stated, draw an arrow iff

Closure(x) \subseteq Closure(y)

. That is, we have an injection from the closure of

x

into the closure of

y

, and the arrow represents the injection. Alternatively, we can think of this as ordering the elements

x

by "information". A point

x

has less information than point

y

if its closure has fewer points.

§ T0 in terms of closure:

$X$ is $T_0$ iff for points $p, q \in X$ , we have an open set $O$ which contains one of the points but not the other. Formally, either $p \in O \land q \not \in O$ , or $p \not \in O \land q \in O$ .
Example of a $T_0$ space is the sierpinski space . Here, we have the open set $\{ \bot \}$ 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 $\bot$ and not $\top$ .
Closure definition: $X$ is $T_0$ iff $x \neq y \implies c(\{x\}) \neq c(\{y\})$

§ T1 in terms of closure:

$X$ is $T_1$ iff for all $p, q$ in $X$ , we have open sets $U_p, U_q$ such that $U_p, U_q$ are open neighbourhoods of $p, q$ which do not contain the "other point". Formally, we need $p \in U_p$ and $q \not \in U_p$ , and similarly $q \in U_q$ and $p \not \in U_q$ . That is, $U_p$ and $U_q$ can split $p, q$ apart, but $U_p$ and $U_q$ need not be disjoint.
Example of $T_1$ is the zariski/co-finite topology on $\mathbb Z$ , where the open sets are complements of finite sets. Given two integers $p, q$ , use the open sets as the complements of the closed finite sets $U_p = \{q\}^C = \mathbb Z - q$ , and $U_q = \{p\}^C = \mathbb Z - p$ . These separate $p$ and $q$ , but have huge intersection: $U_p \cap U_q = Z - \{ p, q\}$ .
Closure definition: $X$ is $T_1$ iff $c(\{x\}) = \{x\}$ .
T1 has all singleton sets as closed

§ Haussdorf (T2) in terms of closure

$X$ ix $T_1$ iff for all $p, q$ in $X$ , we have open set $U_p, U_q$ such that they are disjoint ( $U_p \cap U_q = \emptyset$ ) and are neighbourhoods of $p$ , $q$ : $p \in U_p$ and $q \in U_q$ .
Example of a $T_2$ space is that of the real line, where any two points $p, q$ can be separated with epsilon balls with centers $p, q$ and radii $(p - q) / 3$ .
Closure definition : $X$ is $T_2$ iff $x \neq y$ implies there is a set $A \in 2^X$ such that $x \not \in c(A) \land y \not \in c(X - A)$ where $X - A$ is the set complement.

§ Relationship between DFS and closure when the topology is $T0$

If the topology is

T0

, then we know that the relation will be a poset, and hence the graph will be a DAG. Thus, whenever we have

x \rightarrow y

, we will get

§ Finite topologies and DFS numbering

§ Closure axioms of topology

§ Slight weakening into Single axiom version

§ Continuous functions

§ Finite topologies as preorders

§ T0 in terms of closure:

§ T1 in terms of closure:

§ Haussdorf (T2) in terms of closure

§ Relationship between DFS and closure when the topology is $T0$

§ DFS: the T0 case

§ DFS: the back edges

§ Finite topologies and DFS numbering

§ Closure axioms of topology

§ Slight weakening into Single axiom version

§ Continuous functions

§ Finite topologies as preorders

§ T0 in terms of closure:

§ T1 in terms of closure:

§ Haussdorf (T2) in terms of closure

§ Relationship between DFS and closure when the topology is T0T0T0

§ DFS: the T0 case

§ DFS: the back edges

§ Relationship between DFS and closure when the topology is $T0$