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After reading the comments on Sigfpe's answer I realized that it would be useful to make a rigorous argument to explain why "rulers" or as like to call them "observable properties" should be open sets. In the process I'd like to explain how we can view general topology as an idealized version of computation by interpreting topological spaces as data types and continuous maps as computable functions.

Computationally an observable property $P$ of a data type $A$ corresponds to a semi-decision procedure. In other words a computable function $\chi_P: A \to Unit$ which returns the unique value $()$ of type $Unit$ if $a \in A$ has the property $P$ and runs forever otherwise. We can interpret $P$ as a subset of $A$ and $\chi_P$ as it's characteristic function. Clearly observable properties pull back under computable functions since if $f:B \to A$ is a computable function $\chi_P \circ f$ is a semi-decision procedure.

Let's translate this into topological language. If we interpret $A$ and $B$ as topological spaces and $f:B \to A$ as a continuous map we have that $f^{-1}(P)$ is observable if $P$ is. Thus it makes sense to interpret observable properties as open sets. We can make this correspondence more precise if we notice that every open set $P$ in $A$ corresponds to a map $\chi_P: A \to \mathbb{S}$ where $\mathbb{S}$ is the Sierpinski space. Thus in our translation $Unit$ corresponds to $\mathbb{S}$, the open point of $\mathbb{S}$ corresponds to $()$, and the closed point $\bot$ of $\mathbb{S}$ corresponds to nontermination.

Now the a question remains: why did we choose to represent observable properties by open sets instead of closed sets? The answer lies in the way observable properties behave under intersection and union. Let $P$ and $Q$ be observable properties. The intersection $P \cap Q$ is an observable property we can write a semi-decision procedure $\chi_{P \cap Q}$ by running $\chi_P$ and $\chi_Q$ in succession. Similarly notice that $P \cup Q$ is observable since we can write a semi-decision procedure $\chi_{P \cup Q}$ that runs $\chi_P$ and $\chi_Q$ in parallel and outputs $()$ if one of $\chi_P$ and $\chi_Q$ does. If you have an infinite number of computers it is clear that you can generalize this construction to an infinite union $\bigcup_{i \in I} P_i$ by running all the $\chi_{P_i}$ in parallel. However this will not work for an infinite intersection $\bigcap_{n \in \mathbb{N}} P_n$ because if $\chi_{P_n}$ takes $n$ seconds to terminate, then even running all the $\chi_{P_n}$ in parallel will not help.

I can't help but list out a few other things to ponder in light of this dictionary:

• A space $X$ is discrete if $=:X\times X \to \mathbb{S}$ is continuous

• A space $X$ is Hausdorff if $\neq: X\times X \to \mathbb{S}$ is continuous

• A space $X$ is compact if the map $\forall_X: (X \to \mathbb{S}) \to \mathbb{S}$ is continuous

• An observable property $P$ of $T$ is decidable if and only if $P$ is clopen

• On a sequential machine we can write a semi-decision procedure for a countable union of observable properties but not an uncountable union. Does this say anything about topology?

Here are some nice references:

• Alex Simpson - Topological Spaces from a Computational Perspective

5 added 349 characters in body

After reading the comments on Sigfpe's answer I realized that it would be useful to make a rigorous argument to explain why "rulers" or as like to call them "observable properties" should be open sets. In the process I'd like to explain how we can view general topology as an idealized version of computation by interpreting topological spaces as data types and continuous maps as computable functions.

Computationally an observable property $P$ of a data type $A$ corresponds to a semi-decision procedure. In other words a computable function $\chi_P: A \to Unit$ which returns the unique value $()$ of type $Unit$ if $a \in A$ has the property $P$ and runs forever otherwise. We can interpret $P$ as a subset of $A$ and $\chi_P$ as it's characteristic function. Clearly observable properties pull back under computable functions since if $f:B \to A$ is a computable function $\chi_P \circ f$ is a semi-decision procedure.

Let's translate this into topological language. If we interpret $A$ and $B$ as topological spaces and $f:B \to A$ as a continuous map we have that $f^{-1}(P)$ is observable if $P$ is. Thus it makes sense to interpret observable properties as open sets. We can make this correspondence more precise if we notice that every open set $P$ in $A$ corresponds to a map $\chi_P: A \to \mathbb{S}$ where $\mathbb{S}$ is the Sierpinski space. Thus in our translation $Unit$ corresponds to $\mathbb{S}$, the open point of $\mathbb{S}$ corresponds to $()$, and the closed point $\bot$ of $\mathbb{S}$ corresponds to nontermination.

Now the question remains why did we choose to represent observable properties by open sets instead of closed sets? The answer lies in the way observable properties behave under intersection and union. Let $P$ and $Q$ be observable properties. The intersection $P \cap Q$ is an observable property we can write a semi-decision procedure $\chi_{P \cap Q}$ by running $\chi_P$ and $\chi_Q$ in succession. Similarly notice that $P \cup Q$ is observable since we can write a semi-decision procedure $\chi_{P \cup Q}$ that runs $\chi_P$ and $\chi_Q$ in parallel and outputs $()$ if one of $\chi_P$ and $\chi_Q$ does. If you have an infinite number of computers it is clear that you can generalize this construction to an infinite union $\bigcup_{i \in I} P_i$ by running all the $\chi_i$ \chi_{P_i}$in parallel. However this will not work for an infinite intersection$\bigcap_{n \in \mathbb{N}} P_n$because if$\chi_n$\chi_{P_n}$ takes $n$ seconds to terminate, then even running all the $\chi_n$ \chi_{P_n}$in parallel will not help. I can't help but list out a few other things to ponder in light of this dictionary: - • A space$X$is discrete if$=:X\times X \to \mathbb{S}$is continuous - • A space$X$is Hausdorff if$!=: \neq: X\times X \to \mathbb{S}$is continuous - • A space$X$is compact if the map$\forall_X: (X \to \mathbb{S}) \to \mathbb{S}$is continuous - • An observable property$P$of$T$is decidable if and only if$P$is clopen - • On a sequential machine we can write a semi-decision procedure for a countable union of observable properties but not an uncountable union. Does this say anything about topology? Here are some nice references: • Alex Simpson - Topological Spaces from a Computational Perspective 4 added 68 characters in body After reading the comments on Sigfpe's answer I realized that it would be useful to make a rigorous argument to explain why "rulers" or as like to call them "observable properties" should be open sets. In the process I'd like to explain how we can view general topology as an idealized version of computation by interpreting topological spaces as data types and continuous maps as computable functions. Computationally an observable property$P$of a data type$A$corresponds to a semi-decision procedure. In other words a computable function$\chi_P: A \to Unit$which returns the unique value$()$of type$Unit$if$a \in A$has the property$P$and runs forever otherwise. We can interpret$P$as a subset of$A$and$\chi_P$as it's characteristic function. Clearly observable properties pull back under computable functions since if$f:B \to A$is a computable function$\chi_P \circ f$is a semi-decision procedure. Let's translate this into topological language. If we interpret$A$and$B$as topological spaces and$f:A f:B \to B$A$ as a continuous map we have that $f^{-1}(P)$ is observable if $P$ is. Thus it makes sense to interpret observable properties as open sets. We can make this correspondence more precise if we notice that every open set $P$ in $A$ corresponds to a map $\chi_P: A \to \mathbb{S}$ where $\mathbb{S}$ is the Sierpinski space. Thus in our translation $Unit$ corresponds to $\mathbb{S}$, the open point of $\mathbb{S}$ corresponds to $()$, and the closed point $\bot$ of $\mathbb{S}$ corresponds to nontermination.

Now the question remains why did we choose to represent observable properties by open sets instead of closed sets? The answer lies in the way observable properties behave under intersection and union. Let $P$ and $Q$ be observable properties. The intersection $P \cap Q$ is an observable property we can write a semi-decision procedure $\chi_{P \cap Q}$ by running $\chi_P$ and $\chi_Q$ in succession. Similarly notice that $P \cup Q$ is observable since we can write a semi-decision procedure $\chi_{P \cup Q}$ that runs $\chi_P$ and $\chi_Q$ in parallel and outputs $()$ if one of $\chi_P$ and $\chi_Q$ does. If you have an infinite number of computers it is clear that you can generalize this construction to an infinite union $\bigcup_{i \in I} P_i$ by running all the $\chi_i$ in parallel. However this will not work for an infinite intersection $\bigcap_{n \in \mathbb{N}} P_n$ because if $\chi_n$ takes $n$ seconds to terminate, then even running all the $\chi_n$ in parallel will not help.

I can't help but list out a few other things to ponder in light of this dictionary:
- A space $X$ is discrete if $=:X\times X \to \mathbb{S}$ is continuous
- A space $X$ is Hausdorff if $!=: X\times X \to \mathbb{S}$ is continuous
- A space $X$ is compact if the map $\forall_X: (X \to \mathbb{S}) \to \mathbb{S}$ is continuous
- An observable property $P$ of $T$ is decidable if and only if $P$ is clopen
- We On a sequential machine we can write a semi-decision procedure for a countable union of observable properties on a sequential machinebut not an uncountable union. Does this say anything about topology?