斯科特连续函数：替代定义

我真的很挣扎于此属性：

令为相干空间，而f ：C l （X ）→ C l （Y ）为单调函数。˚F是连续的，当且仅当˚F （⋃ X ∈ d X ）= ⋃ X ∈ d ˚F （X ），对于所有的，使得是一个有向集。$X,Y$$f: Cl(X) \rightarrow Cl(Y)$$f$$f(\bigcup_{x\in D} x)=\bigcup_{x \in D}f(x)$$D \subseteq Cl(X)$$D$

向集合被这样定义：偏序是一个有向组当且仅当这样和。Ç 升（X ）表示X的派系：{ X ＆SubsetEqual; | X | | 一，b ∈ X ⇒ 一个连贯b }。$D \subseteq$$\forall x, x' \in D$ $\exists z \in D$$x \subseteq z$$x' \subseteq z$
$Cl(X)$$\{x \subseteq |X| \mid a,b \in x \Rightarrow a$$b \}$

许多书都将其定义为Scott连续函数的定义，但不幸的是，不是我的老师。他给了我们连续的定义：

是连续当且仅当它是单调和 ∀ X ∈ Ç 升（X ），∀ b ∈ ˚F （X ），∃ X 0 ⊆ ˚F 我Ñ X ，b ∈ ˚F （X 0）， 其中单调定义为： ˚F是单调的当且仅当一个$f : Cl(X) \rightarrow Cl(Y)$$\forall x \in Cl(X), \forall b \in f(x), \exists x_0 \subseteq_{fin} x, b \in f(x_0)$
$f$$a \subseteq b \Rightarrow f(a) \subseteq f(b)$

这是我提出的证明，但我不明白最后一个等式。

证明连续意味着˚F （⋃ d ）= ⋃ ˚F （d $f$$f(\bigcup D)=\bigcup f(D)$：
设。由连续性的定义，∃ X 0 ⊂ ˚F 我Ñ X | b ∈ ˚F （X 0）。请注意，X 0是联盟 { X 我 | X 我 ∈ d }。$b \in f(\bigcup D)$$\exists x_0 \subset_{fin} x \mid b \in f(x_0)$$x_0$$\{ x_i \mid x_i \in D\}$
如果是直接的，那么：∃ ž ∈ d | X 我 ⊆ ž然后X 0 ⊆ ž。由单调的定义，˚F （X 0）⊆ ˚F （ż ）这样b ∈ ˚F （ż ）（???）⊆ ⋃ ˚F （d ）。即使这是真的，我们应该显示⋃ ˚F （d ）= ˚F （⋃ $D$$\exists z \in D \mid x_i \subseteq z$$x_0 \subseteq z$$f(x_0)\subseteq f(z)$$b \in f(z)$ $\subseteq \bigcup f(D)$，而不仅仅是 ⊆。$\bigcup f(D) = f(\bigcup D)$$\subseteq$

其他含义的证明甚至更糟，所以我不能在这里写...您能向我解释证明如何工作吗？

The definition of continuity used by your teacher is the nicer one. It tells you pretty concretely what continuity means.

Suppose $b \in f(x)$. That means that given all the information of $x$, possibly an infinite set of tokens (atoms), the function produces some element that has the atomic piece of information $b$. (It could have other information too, but we are not concerned with that at the moment.) Your teacher's definition says that it is not necessary to look at all the infinite information of $x$ in order to produce the output information $b$. Some finite subset of $x$ is enough to produce it.

（Melvin Fitting的书“可计算性理论，语义和逻辑编程”，牛津，1987年，将此特性称为紧凑性，并将连续函数定义为单调和紧凑型。）

这是连续性的本质。要获得有关函数输出的有限信息，您只需要有关输入的有限信息。通过将函数为无限输入的所有有限近似值生成的信息拼凑在一起，可以获得函数为无限输入产生的输出。换句话说，从有限逼近到它们的无限并集，不会有任何神奇的跳跃。无论您获得无限远，您都应该已经处于有限的阶段。

The standard equation $f(\bigcup_{x \in D} x) = \bigcup_{x \in D} f(x)$ is pretty to look at, but it doesn't tell you all the intuition I have explained above. However, mathematically, it is equivalent to your teacher's definition.

$\bigcup_{x \in D} f(x) \subseteq f(\bigcup_{x \in D} x)$, it is enough to show that $f(x)$ is included in $f(\bigcup_{x \in D} x)$, for each $x \in D$. But that follows directly from monotonicity of $f$ because $x \subseteq \bigcup_{x \in D} x$. So, this is the "easy" direction.

$f(\bigcup_{x \in D} x) \subseteq \bigcup_{x \in D} f(x)$. To see this, use the intuition I have mentioned above. Any atomic piece of information $b$ in the left hand side comes from some finite approximation of the input: $x_0 \subseteq_{fin} \bigcup_{x \in D} x$. That is, $b \in f(x_0)$. Since $x_0$$x_0$$x_0$ itself. Call that element $z$. By monotonicity, $f(x_0) \subseteq f(z)$. So, $b \in f(z)$. Since $z \in D$, $f(z) \subseteq \bigcup_{x \in D} f(x)$. So, now $b$ is seen to be in the right hand side too. QED.

As you have noted, showing that your teacher's continuity implies the pretty equation is the easy bit. The harder bit is to show that the pretty equation, despite looking like it is not saying very much, really does say everything in your teacher's definition.

It occurred to me belatedly, after I wrote the last response, that the teacher's definition of continuity that I was explaining in my response is the topological notion of continuity. The algebraic formulation of continuity that is usually mentioned in Computer Science text books hides all the topological intuitions. (In fact, Dana Scott has often written that he has deliberately avoided topological formulations because Computer Scientists are not familiar with it.)

The linkage between the algebraic and topological formulations is called Stone duality, and it is now becoming increasingly clear that this linkage itself is extremely important for Computer Science.

For a conceptual exposition of these connections (and a lot more), See Abramsky's Information, processes and games.