- Pointclass
In the mathematical field of

descriptive set theory , a**pointclass**is a collection of sets of points, where a "point" is ordinarily understood to be an element of some perfectPolish space . In practice, a pointclass is usually characterized by some sort of "definability property"; for example, the collection of allopen set s in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.)Pointclasses find application in formulating many important principles and theorems from

set theory andreal analysis . Strong set-theoretic principles may be stated in terms of thedeterminacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability (and indeed universal measurability), theproperty of Baire , and theperfect set property .**Basic framework**In practice, descriptive set theorists often simplify matters by working in a fixed Polish space such as Baire space or sometimes

Cantor space , each of which has the advantage of beingzero dimensional , and indeedhomeomorphic to its finite or countable powers, so that considerations of dimensionality never arise. Moschovakis provides greater generality by fixing once and for all a collection of underlying Polish spaces, including the set of all naturals, the set of all reals, Baire space, and Cantor space, and otherwise allowing the reader to throw in any desired perfect Polish space. Then he defines a "product space" to be any finiteCartesian product of these underlying spaces. Then, for example, the pointclass $\backslash boldsymbol\{Sigma\}^0\_1$ of all open sets means the collection of all open subsets of one of these product spaces. This approach prevents $\backslash boldsymbol\{Sigma\}^0\_1$ from being aproper class , while avoiding excessive specificity as to the particular Polish spaces being considered (given that the focus is on the fact that $\backslash boldsymbol\{Sigma\}^0\_1$ is the collection of open sets, not on the spaces themselves).**Boldface pointclasses**The pointclasses in the

Borel hierarchy , and in the more complexprojective hierarchy , are represented by sub- and super-scripted Greek letters inboldface fonts; for example, $\backslash boldsymbol\{Pi\}^0\_1$ is the pointclass of allclosed set s, $\backslash boldsymbol\{Sigma\}^0\_2$ is the pointclass of all "F"_{σ}sets, $\backslash boldsymbol\{Delta\}^0\_2$ is the collection of all sets that are simultaneously "F"_{σ}and "G"_{δ}, and $\backslash boldsymbol\{Sigma\}^1\_1$ is the pointclass of allanalytic set s.Sets in such pointclasses need be "definable" only up to a point. For example, every

singleton set in a Polish space is closed, and thus $\backslash boldsymbol\{Pi\}^0\_1$. Therefore it cannot be that every $\backslash boldsymbol\{Pi\}^0\_1$ set must be "more definable" than an arbitrary element of a Polish space (say, an arbitrary real number, or an arbitrary countable sequence of natural numbers). Boldface pointclasses, however, may (and in practice ordinarily do) require that sets in the class be definable relative to some real number, taken as an oracle. In that sense, membership in a boldface pointclass is a definability property, even though it is not absolute definability, but only definability with respect to a possibly undefinable real number.Boldface pointclasses, or at least the ones ordinarily considered, are closed under

Wadge reducibility ; that is, given a set in the pointclass, itsinverse image under acontinuous function (from a product space to the space of which the given set is a subset) is also in the given pointclass. Thus a boldface pointclass is a downward-closed union ofWadge degree s.**Lightface pointclasses**The Borel and projective hierarchies have analogs in

effective descriptive set theory in which the definability property is no longer relativized to an oracle, but is made absolute. For example, if one fixes some collection of basic open neighborhoods (say, in Baire space, the set of all sets of the form {"x"∈ω^{ω}|"x" ⊇"s"} for any fixed finite sequence "s" of natural numbers), then the open, or $\backslash boldsymbol\{Sigma\}^0\_1$, sets may be characterized as all (arbitrary) unions of basic open neighborhoods. The analogous $Sigma^0\_1$ sets, with a lightface $Sigma$, are no longer "arbitrary" unions of such neighborhoods, but computable unions of them (that is, a set is $Sigma^0\_1$ if there is a computable set "S" of finite sequences of naturals such that the given set is the union of all {"x"∈ω^{ω}|"x" ⊇"s"} for "s" in "S"). A set is lightface $Pi^0\_1$ if it is the complement of a $Sigma^0\_1$ set. Thus each $Sigma^0\_1$ set has at least one**index**, which describes the computable function enumerating the basic open sets from which it is composed; in fact it will have infinitely many such indices. Similarly, an index for a $Pi^0\_1$ set "B" describes the computable function enumerating the basic open sets in the complement of "B".A set "A" is lightface $Sigma^0\_2$ if it is a union of a computable sequence of $Pi^0\_1$ sets (that is, there is a computable enumeration of indices of $Pi^0\_1$ sets such that "A" is the union of these sets). This relationship between lightface sets and their indices is used to extend the lightface Borel hierarchy into the transfinite, via

recursive ordinal s. This produces thathyperarithmetic hierarchy , which is the lightface analog of the Borel hierarchy. (The finite levels of the hyperarithmetic hierarchy are known as thearithmetical hierarchy .)A similar treatment can be applied to the projective hierarchy. Its lightface analog is known as the

analytical hierarchy .**References***

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