- We can see that the cardinality of $\mathbb R$ is different from the cardinality of the space of functions over it, $\mathbb R \rightarrow \mathbb R$.
- However, "the set of all functions" isn't really something mathematicians consider. One would most likely consider "the set of all
*continuous*functions" $\mathbb R \rightarrow \mathbb R$. - Now note that a function that is continuous over the reals is determined by its values at the rationals . So, rather than giving me a continus function $f: \mathbb R \rightarrow \mathbb R$, you can give me a continuous function $f': \mathbb Q \rightarrow \mathbb R$ which I can Cauchy-complete, to get a function $\texttt{completion}(f') : \mathbb R \rightarrow \mathbb R = f$.
- Now, cardinality considerations tell us that:

$|\mathbb R^\mathbb Q| = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = 2^\aleph_0 = |R|$

- We've won! We have a space $\mathbb R$ whose space of
*continuous*functions $\mathbb R \rightarrow \mathbb R$ is isomorphic to $\mathbb R$. - We bravely posit: all functions computed by lambda-calculus are continuous! Very well. This leaves us two questions to answer to answer: (1) over what space? (2) with what topology? The answers are (1) a space of partial orders (2) with the Scott topology

- A DCPO (directed-complete partial order) is an algebraic structure that can be satisfied by some partial orders. This definition ports 'continuity' to partial orders.

- A domain is an algebraic structure of even greater generality than a DCPO. This attempts to capture the fundamental notion of 'finitely approximable'.

- The presentation of a domain is quite messy. The nicest axiomatization of domains that I know of is in terms of information systems . One can find an introduction to these in the excellent book 'Introduction to Order Theory' by Davey and Priestly

- Given a partial order $(P, \leq)$. assume we have a subset $Q \subseteq P$. A least upper bound $u$ of $Q$ is an element that is the smallest element in $P$which is larger than every element in $Q$.

- A subset $Q$ of $P$ is called as a chain if its elements can be put into order. That is, there is a labelling of elements of $Q$ into $q1, q2, \dots, qn$ such that $q1 \leq q2 \leq \dots \leq qn$.

- A partially ordered set is called as a
*chain complete partial order*if each chain has a least upper bound.

- This is different from a lattice, where each
*subset*has a least upper bound.

- Every ccpo has a minimal element given by $completion(\emptyset) = \bot$.

- TODO: example of ccpo that is not a lattice

- A function from $P$ to $Q$ is said to be monotone if $p \leq p' \implies f(p) \leq f(p')$.
- Composition of monotone functions is monotone.
- The image of a chain wrt a monotone function is a chain.
- A monotone function
. Consider:*need not preserve least upper bounds*

$f: 2^{\mathbb N} \rightarrow 2^{\mathbb N}
f(S) \equiv
\begin{cases}
S & \text{$S$} is finite \\
S U \{ 0 \} &\text{$S$ is infinite}
\end{cases}$

This does not preserve least-upper-bounds. Consider the sequence of elements:
$A_1 = \{ 1\}, A_2 = \{1, 2\}, A_3 = \{1, 2, 3\}, \dots, A_n = \{1, 2, 3, \dots, n \}$

The union of all $A_i$ is $\mathbb N$.
Each of these sets is finite.
Hence $f(\{1 \}) = \{1 \}$, $f(\{1, 2 \}) = \{1, 2\}$ and so on. Therefore:
$f(\sqcup A_i) = f(\mathbb N) = \mathbb N \cup \{ 0 \}\\
\sqcup f(A_i) = \sqcup A_i = \mathbb N$

- A function is continous if it is monotone and preserves all LUBs. This is only sensible as a definition on ccpos, because the equation defining it is:
`lub . f = f . lub`

, where`lub: chain(P) \rightarrow P`

. However, for`lub`

to always exist, we need`P`

to be a CCPO. So, the definition of continuous only works for CCPOs. - The composition of continuous functions of chain-complete partially ordered sets is continuous.

$\texttt{FIX}(f) \equiv \texttt{lub}(\{ f^n(\bot) : n \geq 0 \})$

- Semantics with Applications: Hanne Riis Nielson, Flemming Nielson.
- Lecture notes on denotational semantics: Part 2 of the computer science Tripos
- Outline of a mathematical theory of computation
- Domain theory and measure theory: Video