Cantor space ($\{0,1{\}}^{\mathbb{N}}$)

Cantor space is the binary sequence space with the product topology of discrete two-point factors. Here the Bernoulli-space synonym refers to this topology, without selecting a particular probability measure. The metric $d(x,y)={\sum }_{j\ge 1}{2}^{-j}|x_j-y_j|$ induces the topology. A diagonal subsequence argument gives compactness. Fixing a finite prefix gives a clopen cylinder set. The map $x\mapsto {\sum }_{j}2x_j{3}^{-j}$ is a homeomorphism to the usual Cantor set.

A binary tree of finite strings is a set of finite binary strings closed under taking prefixes. The empty tree is allowed; a nonempty such tree contains the empty string. Finite nodes may have no infinite extensions. Such presentations describe closed subsets of Cantor space through their infinite paths.

A binary tree of finite strings is computable when membership of a finite binary string is decidable. An infinite computable tree need not have any computable infinite path. Removing all nodes without infinite extensions need not preserve decidability.

For an effective enumeration $({\phi }_e)$ of unary partial computable functions, admit a length-$n$ string $\sigma$ if it disagrees with every binary value of ${\phi }_e(e)$ observed within $n$ steps for $e<n$. The bounded halting predicate makes membership decidable, and the tests are prefix compatible. Its paths are exactly the binary diagonally noncomputable functions. The path space is nonempty and perfect because infinitely many indices of divergent programs leave arbitrarily late bits free; no path is computable. Dead ends in this decidable presentation are essential.

For a nonempty decidable binary tree of finite strings with no terminal nodes, begin at the empty string and repeatedly choose the first immediate successor that belongs to the tree. Each finite decision terminates and some successor always exists. This produces a total computable function giving an infinite path. In particular, a nonempty decidable tree in which every node has two incompatible extensions cannot have only noncomputable paths.

An infinite path is a binary function all of whose finite prefixes belong to the tree. Its path space $[T]$ is closed in Cantor space, since failure is witnessed by one finite prefix outside $T$.

A nonempty path space $[T]$ is perfect if it has no isolated points: for every path and every finite prefix of it there is a different path sharing that prefix. This property of the closed subset of Cantor space does not imply that its particular finite-string presentation has no dead ends. Requiring every finite node to extend to incompatible nodes is the stronger pruned-tree convention for a perfect tree.

Every open set in Cantor space is a countable disjoint union of prefix cylinder sets. Select all finite words whose cylinder lies in the open set but whose immediate predecessor's cylinder does not. These words are prefix-free, so their cylinders are disjoint. Every point of the open set has a shortest qualifying prefix. The whole space is represented by the empty prefix.

A continuous function on Cantor space is uniformly continuous. Replacing it by one constant on each length-$n$ prefix cylinder set changes it by at most its oscillation on sets of diameter $2^{-n}$. This tends uniformly to zero. Each approximant is a finite linear combination of continuous indicator functions, proving density in the supremum norm.