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Fσ set

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In mathematics, an F_σ set (pronounced F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme (French: sum, union).^[1]

The complement of an F_σ set is a G_δ set.^[1]

F_σ is the same as $\mathbf {\Sigma } _{2}^{0}$ in the Borel hierarchy.

Examples

[edit]

Each closed set is an F_σ set.

The set $\mathbb {Q}$ of rationals is an F_σ set in $\mathbb {R}$ . More generally, any countable set in a T₁ space is an F_σ set, because every singleton $\{x\}$ is closed.

The set $\mathbb {R} \setminus \mathbb {Q}$ of irrationals is not an F_σ set.

In metrizable spaces, every open set is an F_σ set.^[2]

The union of countably many F_σ sets is an F_σ set, and the intersection of finitely many F_σ sets is an F_σ set.

The set $A$ of all points $(x,y)$ in the Cartesian plane such that $x/y$ is rational is an F_σ set because it can be expressed as the union of all the lines passing through the origin with rational slope:

A=\bigcup _{r\in \mathbb {Q} }\{(ry,y)\mid y\in \mathbb {R} \},

where $\mathbb {Q}$ is the set of rational numbers, which is a countable set.

References

[edit]

^ ^a ^b Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560.
^ Aliprantis, Charalambos D.; Border, Kim (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, p. 138, ISBN 9783540295877.

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In mathematics, particularly within the fields of general topology and descriptive set theory, an F_σ set is a subset of a topological space that can be expressed as a countable union of closed sets.^[1] The notation "F_σ" originates from French mathematical terminology, where "F" stands for fermé (closed) and "σ" denotes a countable union (from "somme," meaning sum).^[1] F_σ sets constitute the second level (denoted Σ²₀) of the Borel hierarchy, a classification system for Borel sets generated iteratively from open sets through countable unions and complements; all F_σ sets are thus Borel sets. This hierarchy, developed in the early 20th century, underpins much of modern analysis by organizing sets based on their constructive complexity from basic open sets.^[2] Key properties include the fact that the complement of an F_σ set is a G_δ set (a countable intersection of open sets), and in metric spaces like the real numbers ℝ, every F_σ set is Lebesgue measurable. Moreover, in ℝ, both open sets and closed sets qualify as F_σ sets: closed sets trivially so, while open sets can be written as countable unions of closed intervals (e.g., an open interval (a, b) equals ⋃_{n=1}^∞ [a + 1/n, b - 1/n]). Notable examples of F_σ sets include all countable subsets of ℝ (as unions of closed singletons), the set of rational numbers ℚ (countable and dense), and the Cantor set (closed, hence F_σ). However, not all Borel sets are F_σ; higher levels of the hierarchy, such as G_δσ sets (countable unions of G_δ sets), extend beyond this class.^[2] F_σ sets are fundamental in measure theory for proving regularity of measures and in descriptive set theory for analyzing Polish spaces, where they help characterize properties like Baire category and comeager sets. Their role extends to applications in functional analysis and probability, ensuring that many naturally occurring sets (e.g., domains of continuous functions) possess desirable structural features.^[3]

Fundamentals

Definition

In a topological space

X

, an F_\sigma set is a countable union of closed subsets of

X

. Formally, a subset

S \subseteq X

is an F_\sigma set if it can be expressed as

S = \bigcup_{n=1}^\infty F_n

, where each

F_n

is a closed subset of

X

.^[4] The term "countable" here refers to a union over at most countably many sets, which includes finite unions as a special case (by taking the remaining

F_n

to be empty for sufficiently large

n

).^[5] Closed sets in a topological space are those whose complements are open sets.^[6]

Notation

The standard notation for a countable union of closed sets in a topological space is an Fσ set, where "F" stands for the French term fermé (closed) and σ symbolizes a countable union, derived from somme (sum).^[1] This convention reflects the French origins of early 20th-century developments in point-set topology and descriptive set theory.^[1] In mathematical literature, the notation is frequently rendered as F_σ, with σ as a subscript, particularly in printed texts and LaTeX typesetting (e.g.,

F_\sigma

). In English-language sources, these sets are termed "F-sigma sets," pronounced as "eff-sigma."^[1] An alternative notation appears in some contexts of descriptive set theory, where Fσ sets correspond to the class

\Sigma^0_2

in the Borel hierarchy (a full explanation is deferred to the Borel hierarchy section).

Properties

Algebraic Properties

F_σ sets exhibit specific behaviors under set-theoretic operations, reflecting their position in the Borel hierarchy as countable unions of closed sets. The collection of F_σ sets is closed under countable unions. Suppose

\{A_k\}_{k=1}^\infty

is a countable family of F_σ sets, where each

A_k = \bigcup_{n=1}^\infty C_{k,n}

with

C_{k,n}

closed in the topological space. Then,

\bigcup_{k=1}^\infty A_k = \bigcup_{k=1}^\infty \bigcup_{n=1}^\infty C_{k,n} = \bigcup_{(k,n) \in \mathbb{N} \times \mathbb{N}} C_{k,n}.

Since

\mathbb{N} \times \mathbb{N}

is countable, this is a countable union of closed sets, hence an F_σ set.^[1] The collection is also closed under finite intersections. For two F_σ sets

A = \bigcup_{m=1}^\infty C_m

and

B = \bigcup_{n=1}^\infty D_n

, where each

C_m

and

D_n

is closed,

A \cap B = \bigcup_{m=1}^\infty \bigcup_{n=1}^\infty (C_m \cap D_n).

The intersection

C_m \cap D_n

is closed, and the double union is countable, so

A \cap B

is F_σ. This property extends to any finite number of F_σ sets by induction.^[7] However, F_σ sets are not closed under arbitrary (possibly uncountable) unions. A counterexample arises in

\mathbb{R}

with the Vitali set

V \subset [0,1]

, constructed using the axiom of choice by selecting one representative from each coset of

\mathbb{R}/\mathbb{Q}

[0,1]

. The set

V

is the union of uncountably many singletons

\{r\}

, each of which is closed (and thus F_σ), but

V

is not Borel measurable and therefore not an F_σ set.^[8] The complement of an F_σ set is a

G_\delta

set. If

F = \bigcup_{n=1}^\infty C_n

with each

C_n

closed, then

X \setminus F = \bigcap_{n=1}^\infty (X \setminus C_n),

where each

X \setminus C_n

is open, yielding a countable intersection of open sets.^[1]

Topological Properties

In topological spaces, Fσ sets form an important subclass of the Borel σ-algebra, which is generated by the open sets. Specifically, every Fσ set, being a countable union of closed sets, belongs to the Borel σ-algebra since closed sets are Borel and the Borel σ-algebra is closed under countable unions.^[9] This inclusion holds in any topological space, ensuring that Fσ sets inherit the measurability properties associated with Borel sets in contexts where a measure is defined on the Borel σ-algebra.^[9] A key topological feature of certain Fσ sets relates to meagerness in the sense of the Baire category theorem. A set is meager (or of first category) if it is a countable union of nowhere dense sets, and when these nowhere dense sets are taken to be closed, the resulting set is precisely an Fσ set that is meager. Nowhere dense closed sets have empty interior, and their countable union forms an Fσ set with empty interior in complete metric spaces without isolated points, by the Baire category theorem. This characterization underscores the role of Fσ sets in distinguishing "small" subsets in terms of topological density. In metric spaces, Fσ sets exhibit additional structure related to compactness. An Fσ set is σ-compact if it can be expressed as a countable union of compact sets, which occurs when each of the closed sets in the union is compact.^[10] Since compact subsets of metric spaces are closed, such Fσ sets are precisely the σ-compact subsets, providing a bridge between sequential compactness and the Fσ class.^[10] Not every open set in a general topological space is an Fσ set, as demonstrated by the co-countable topology on an uncountable set X, where open sets are the empty set and complements of countable subsets. In this topology, closed sets are the countable subsets and X itself; thus, any non-empty open set, being uncountable, cannot be a countable union of countable closed sets, hence not Fσ. In contrast, every closed set is trivially an Fσ set, as it is the union of a single closed set.^[4]

Hierarchy and Relations

Borel Hierarchy

The Borel hierarchy provides a systematic classification of Borel sets within a topological space, constructed iteratively from the open sets through transfinite applications of countable unions and intersections. It consists of two dual sequences of classes, denoted

\Sigma^0_\alpha

and

\Pi^0_\alpha

for countable ordinals

\alpha

, where

\Sigma^0_1

comprises the open sets and

\Pi^0_1

the closed sets. Subsequent levels are defined recursively: for

\alpha > 1

, a set belongs to

\Sigma^0_\alpha

if it is a countable union of sets from some

\Pi^0_\beta

with

\beta < \alpha

, while

\Pi^0_\alpha

consists of countable intersections of sets from

\Sigma^0_\beta

for

\beta < \alpha

. This hierarchy stratifies the Borel

\sigma

-algebra, which is the smallest

\sigma

-algebra containing the open sets, and its length is

\omega_1

, the first uncountable ordinal, in spaces like the reals.^[11]^[12] F

\sigma

sets occupy the

\Sigma^0_2

level of this hierarchy, defined as countable unions of closed sets (i.e., sets in

\Pi^0_1

). Thus, every F

\sigma

set can be expressed as

\bigcup_{n=1}^\infty F_n

where each

F_n

is closed, and conversely, in standard settings, every

\Sigma^0_2

set arises this way. This class includes all closed sets (as singleton unions) and extends to finite unions of closed sets, but the hierarchy's strictness ensures that not all

\Sigma^0_2

sets reduce to lower levels. The progression continues with higher finite levels, such as

\Pi^0_3

, which includes F

\sigma\delta

sets—countable intersections of F

\sigma

sets—properly containing the F

\sigma

class.^[11]^[12] In Polish spaces—separable completely metrizable topological spaces—the

\Sigma^0_2

class is precisely the collection of countable unions of closed sets, aligning exactly with F

\sigma

sets and facilitating effective descriptive set-theoretic analysis. The strict inclusion of levels, first established by Lebesgue in 1905 via diagonalization arguments, confirms that closed sets (

\Pi^0_1

) form a proper subclass of F

\sigma

sets, while F

\sigma

sets are properly contained within more complex Borel classes like

\Pi^0_3

.^[11]^[12]

Duality with Gδ Sets

Gδ sets are defined as countable intersections of open sets in a topological space and correspond to the class

\Pi^0_2

in the Borel hierarchy.^[13] This structure positions them as the dual counterparts to Fσ sets within the descriptive hierarchy of Borel sets. The duality between Fσ and Gδ sets arises from De Morgan's laws applied to countable operations. Specifically, the complement of an Fσ set, expressed as a countable union of closed sets

A = \bigcup_{n=1}^\infty F_n

where each

F_n

is closed, is a Gδ set given by

A^c = \bigcap_{n=1}^\infty F_n^c

, with each

F_n^c

open. Conversely, the complement of a Gδ set is an Fσ set.^[14] This complementary relationship underscores their symmetric roles in topological constructions. In general, Fσ sets and Gδ sets are not equivalent classes. For instance, the set of rational numbers ℚ in the real line ℝ is an Fσ set, as it is a countable union of closed singletons, but it is not a Gδ set; if it were, its complement (the irrationals) would be Fσ, leading to a contradiction with the Baire category theorem, which asserts that ℝ cannot be a countable union of nowhere dense closed sets.^[14] In complete metric spaces, zero sets of continuous real-valued functions—defined as

Z(f) = \{ x \mid f(x) = 0 \}

—are precisely the closed Gδ sets.^[15]

Examples

In the Real Line

In the real line

\mathbb{R}

with the standard Euclidean topology,

F_\sigma

sets arise frequently as countable unions of closed subsets, providing intuitive illustrations of this class of sets.^[1] The set of rational numbers

\mathbb{Q}

serves as a classic example of an

F_\sigma

set that is neither open nor closed. Since

\mathbb{Q}

is countable, enumerate its elements as

\{q_n : n \in \mathbb{N}\}

; then

\mathbb{Q} = \bigcup_{n=1}^\infty \{q_n\}

, where each singleton

\{q_n\}

is closed in

\mathbb{R}

. Similarly, the set of integers

\mathbb{Z}

is an

F_\sigma

set, as it is a countable (in fact, finite in any bounded interval) union of closed singletons, and moreover

\mathbb{Z}

is itself closed in

\mathbb{R}

.^[16] The Cantor set

\mathcal{C} \subset [0,1]

, constructed by iteratively removing middle-third open intervals from

[0,1]

, is a canonical closed uncountable set of Lebesgue measure zero, hence an

F_\sigma

set as the union of a single closed set (itself).^[17] Its complement

[0,1] \setminus \mathcal{C}

is open, and thus both an

F_\sigma

set and a

G_\delta

set in

\mathbb{R}

.^[1] Half-open intervals provide another representative example of

F_\sigma

sets that are neither closed nor open. For instance, the interval

(0,1] = \bigcup_{n=1}^\infty \left[\frac{1}{n}, 1\right]

, where each

\left[\frac{1}{n}, 1\right]

is closed in

\mathbb{R}

.^[1] This construction highlights how

F_\sigma

sets can approximate open sets through exhaustion by closed subsets.

In General Topological Spaces

In the discrete topology on a set

X

, every subset is both open and closed. Consequently, every subset of

X

is an Fσ set, as it can be expressed as the union of a single closed set—namely, itself—regardless of whether

X

is countable or uncountable. The Sierpiński space provides a simple illustration of Fσ sets in a non-Hausdorff topology. This space consists of the set

\{0,1\}

with open sets

\emptyset

\{0\}

, and

\{0,1\}

, making the closed sets

\emptyset

\{1\}

, and

\{0,1\}

. The Fσ sets are then the countable unions of these closed sets; since the space is finite, these coincide with the finite unions, yielding

\emptyset

\{1\}

, and

\{0,1\}

. For instance,

\{1\}

is an Fσ set as a single closed set, but the open set

\{0\}

is not Fσ, as it cannot be obtained as such a union. In the cofinite topology on an uncountable set

X

, the closed sets comprise all finite subsets of

X

and

X

itself. Thus, the Fσ sets consist of all countable subsets of

X

(as countable unions of finite closed sets) and

X

(as a single closed set). However, a proper uncountable subset whose complement is countably infinite—known as a proper co-countable set—is not Fσ, since it is neither countable nor equal to

X

, and cannot be formed as a countable union of finite closed sets without including

X

. This serves as a counterexample showing that not all uncountable subsets are Fσ in general topological spaces.

Applications

In Real Analysis

In real analysis, Fσ sets play a fundamental role due to their membership in the Borel σ-algebra, which ensures they inherit key properties related to measurability and approximation. Specifically, every Fσ set in

\mathbb{R}^n

is a Borel set, as it arises from countable unions of closed sets generated from the open sets via the Borel hierarchy. Consequently, all Fσ sets are Lebesgue measurable, since the Borel σ-algebra is contained within the Lebesgue σ-algebra on

\mathbb{R}^n

. This measurability implies that Fσ sets admit well-defined Lebesgue measures, facilitating their use in integration theory and approximation arguments. A notable application involves the structure of discontinuities for real-valued functions on

\mathbb{R}

. For any function

f: \mathbb{R} \to \mathbb{R}

, the set of points of continuity forms a

G_\delta

set, while the set of discontinuities

D_f = \{x \in \mathbb{R} : f \text{ is discontinuous at } x\}

is an Fσ set.^[18] This follows from expressing

D_f

as the countable union

\bigcup_{n=1}^\infty D_{1/n}

, where each

D_\epsilon = \{x \in \mathbb{R} : \omega_f(x) \geq \epsilon\}

is closed, with

\omega_f(x)

denoting the oscillation of

f

x

.^[18] For regulated functions, such as those of bounded variation, the discontinuities are at most countable and thus form an Fσ set of measure zero, enhancing their analytical tractability. In

\mathbb{R}^n

, σ-compact sets—defined as countable unions of compact subsets—are precisely the Fσ sets that are σ-finite with respect to Lebesgue measure, as each compact subset has finite measure and closed topology. Such sets often arise in contexts requiring controlled growth, like supports of test functions in distribution theory, where their finite-measure components allow for effective integration bounds. However, σ-compact sets need not have finite total measure, as exemplified by

\mathbb{R}^n

itself, though they do in bounded domains or when restricted to regions of interest.^[19] Lusin's theorem further underscores the utility of Fσ sets by linking measurability to near-continuity. For a Lebesgue measurable function

f: \mathbb{R} \to \mathbb{R}

, given

\epsilon > 0

, there exists a continuous function

g: \mathbb{R} \to \mathbb{R}

such that

f

and

g

agree except on an Fσ set of measure less than

\epsilon

.^[19] This Fσ set can be taken σ-compact, reflecting the theorem's proof via exhaustion of

\mathbb{R}

by compacta and approximation on each, thereby ensuring

f

behaves continuously outside a controlled exceptional set of small measure.^[19]

In Descriptive Set Theory

In descriptive set theory, Fσ sets are studied primarily within Polish spaces, where they form an important subclass of Borel sets due to their low complexity in the Borel hierarchy. Specifically, an Fσ set in a Polish space

X

is a

\Sigma^2_0

set, meaning it belongs to the second level of the additive Borel hierarchy as a countable union of closed sets.^[20] While Fσ sets are definable at higher levels in the projective hierarchy—namely, as

\Delta^1_1

sets, being both analytic and co-analytic—their primary significance lies in this Borel position, which ensures strong regularity properties without invoking projective determinacy or other axioms.^[20] Fσ sets are analytic sets, or

\Sigma^1_1

sets in the projective hierarchy, as they arise as continuous images of closed subsets of Polish spaces. In particular, each closed component of an Fσ set is the continuous image of itself (a Polish space), and the countable union structure allows embedding into a closed subset of the Baire space

\mathbb{N}^\mathbb{N}

via a continuous surjection.^[20] This places Fσ sets among the Souslin sets, which coincide with analytic sets in Polish spaces, highlighting their role in bridging Borel and projective definability. A key theorem concerning Fσ sets is the perfect set property: every uncountable Fσ subset of a Polish space contains a perfect subset, and thus has cardinality

2^{\aleph_0}

.^[21] In the real line

\mathbb{R}

, this implies that any non-empty perfect Fσ set has the cardinality of the continuum, as perfect sets in

\mathbb{R}

are uncountable and embed the Cantor set.^[21] Regarding relations, an Fσ relation on Polish spaces—viewed as an Fσ subset of the product space—is Borel and thus analytic. Uniformization of such relations by Borel selectors is possible in ZFC when the vertical sections are countable, via the Lusin–Novikov theorem, which guarantees a uniformization consisting of countably many Borel functions. For general Fσ relations without this section condition, Borel uniformization requires additional axioms such as the axiom of determinacy for projective sets or Martin's axiom for countable posets.^[22]

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Fσ set

Fundamentals

Definition

Notation

Properties

Algebraic Properties

Topological Properties

Hierarchy and Relations

Borel Hierarchy

Duality with Gδ Sets

Examples

In the Real Line

In General Topological Spaces

Applications

In Real Analysis

In Descriptive Set Theory

References

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Fσ set

Examples

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Fσ set

Fundamentals

Definition

Notation

Properties

Algebraic Properties

Topological Properties

Hierarchy and Relations

Borel Hierarchy

Duality with Gδ Sets

Examples

In the Real Line

In General Topological Spaces

Applications

In Real Analysis

In Descriptive Set Theory

References

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