We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb N}}$ , (...) there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

A set Asubset of or equal toω is called computably enumerable , if there is an algorithm to enumerate the elements of it. For sets A,Bsubset of or equal toω, we say that A is bounded Turing reducible to reducible to) B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B, written image. Let image be the structure of the c.e. bT-degrees, (...) the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in image. We show that for any c.e. bT-degree image, there is a c.e. bT-degree image such that for any c.e. bT-degree image, image if and only if image. We prove that the analog of the Seetapun local noncappability theorem from the c.e. Turing degrees also holds in image. This theorem demonstrates that every image is noncappable with any nontrivial degree below some image. (shrink)

An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes and (...) for the subclasses of decidable and of homogeneous ${\Pi^{0}_{1}}$ classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. (shrink)