Open access peer-reviewed chapter

Covers and Properties of Families of Real Functions

Written By

Lev Bukovský

Reviewed: 21 September 2021 Published: 24 June 2022

DOI: 10.5772/intechopen.100555

From the Edited Volume

Advanced Topics of Topology

Edited by Francisco Bulnes

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Abstract

We present results on the relationships of the covering property GΦΨ for Φ,Ψ∈OΛΩΓ and G∈S1SfinUfin of a topological space and the selection property GΦ0Ψ0 of the corresponding family of real functions. The result already published are presented without a proof, however with a citation of the corresponding paper. We present a general Theorem that covers almost all the result of this kind. Some results about hereditary properties are enclosed. We also present Scheepers Diagram of considered covering properties for uncountable covers.

Keywords

  • covering properties S1
  • Sfin
  • Ufin
  • selection principles S1
  • Sfin
  • Ufin∗
  • Scheepers Diagram
  • A-measurable function
  • upper A-semimeasurable function
  • hereditary properties
  • σ-space

1. Introduction

The paper is a collection of several results concerning the equivalences of the covering properties of a topological space X and the properties of the family of real functions defined on X and related to this cover. Indeed, we shall present results about the equivalences of the covering properties GΦΨ of a topological space X and the selection property GΦ0Ψ0 of the topological space of upper semicontinuous real functions USCpX on X, for Φ,ΨOΛΩΓ and GS1SfinUfin. The upper semicontinuous functions in this connection were for first time used in [1]. In some important cases we can replace the space USCpX by the topological space CpX of continuous functions. So we obtain the equivalence of some topological property of CpX and a covering property of X.

It turned out that we can prove a general theorem about measurable covers in a very abstract sense that covers almost all the special results.

The covering properties GΦΨ were essentially introduced by M. Scheepers [2] and then, for countable covers, systematically investigated by W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki [3]. Using proved equivalences they obtained the quite simple Scheepers Diagram for Countable Covers. However, not all equivalences are true for arbitrary covers, therefore the corresponding Scheepers Diagram for Arbitrary Infinite Covers, presented below, is more complicated.

Then we present some results about hereditary properties of considered covering properties for Fσ-subsets. The results are important in many considerations. Finally, inspired by the result of J. Haleš [4], we show some relationships of the hereditary property of the topological space X for any subset with the property being a σ-space.

If the presented result was already published, we present the precise citation and no proof.

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2. Notations and terminology

By a topological space we understand an infinite Hausdorff space Xτ, where τ is the topology on X: the set of all open subsets of X [5]. The smallest σ-algebra containing all open sets is the family BORELX=BOREL of Borel subsets of X.

We recall some covering properties which were introduced by M. Scheepers in [2]. If A,BPY are sets of subsets of a set Y, then the covering principle S1AB means the following: for every sequence Un:nω of elements of A and for every nω there exists a UnUn such that Un:nωB. The covering principles SfinAB and UfinAB are there defined in a similar way. In the case of SfinAB we choose finite sets WnUn such that nWnB. In the case of SfinAB we chose finite sets WnUn such that nWnB. In the case of UfinAB we chose finite sets WnUn such that Wn:nωB. Actually we tacitly assume that Y=PX. If YXR we define the selection property UfinAB similarly, but we ask1 that minWn:nωB.

A family U of subsets of X is a cover of X if U=X. A cover is open, if every element of the cover is an open set. A cover VU is said to be a subcover of the cover U. If we deal with a countable cover of X we can consider it as a sequence of subsets.

If A is a family of subsets of X, then we denote by Actbl the family of all countable elements of A. If AXR then A+ is the family fA:xXfx0.

We introduce four special types of covers of a given set. From some technical reasons a cover is said to be an o-cover. A cover U of a set X is a λ-cover if for every xX the set UU:xU is infinite. A cover U of a set X is an ω-cover if XU and for every finite FX there exists a UU such that FU. Finally, a cover U of a set X is a γ-cover if U is infinite and for every xX the set UU:xU is finite.

We shall use the following convention. If the lower case letters φ or ψ denote one of the symbols o, λ, ω or γ, then the capital letters Φ or Ψ denote the corresponding symbol O, Λ, Ω or Γ, respectively, and vice versa.

Assume that EPX and ,XE. Dealing with the covering property Ufin we assume that E is closed under finite unions.

Let φoλωγ. We denote by ΦE the family of all φ-covers U of X satisfying UE. If Xτ is an infinite Hausdorff topological space, then Φτ is simply denoted as Φ.

Evidently (we should eventually omit X from a γ-cover)

ΓEΩEΛEOE.E1

We say that a family VPX is a refinement of the family UPX if

VVUUVU.E2

Let the family VPX be a refinement of the family UPX. If V is an o- or an ω-cover, then U is such a cover as well. This is not true for λ- and γ-covers. If we add finitely many subsets of X to a φ-cover, φoλωγ, we obtain a φ-cover. Any infinite subset of a γ-cover is a γ-cover as well.

A set ωR of all sequences of reals is quasi-ordered by the eventual dominating relation

φψn0ωnn0φnψn.E3

A set FωR is called bounded if there exists a sequence ψωR such that φψ for every φF. The set F is dominating if for every ψωR there exists a φF such that ψφ. The cardinals b and d (see e.g. [6, 7]) are the smallest cardinalities of an unbounded and dominating family, respectively.

The set XR is endowed with the product topology. For any real aR we denote by a the constant function defined on X with value a. There are at least two important subfamilies of XR: the family CX of all continuous functions and the family USCX of all upper semicontinuous functions2. If they are endowed with the product topology we write CpX and USCpX, respectively. Note that fnf in the product topology if and only if xXfnxfx.

We introduce three properties of an infinite family FXR of real functions.

O00x=0fx:fF¯foreveryxX.Ω00Fand0F¯intheprodiucttopologyofXR.Γ0Fisinfiniteandforeveryε>0andforeveryxXthesetfF:fxεisfinite.

Let Φ and Ψ be one of the symbols O,Ω,Γ. Similarly as for covers, we define for an infinite family FXR of real functions the set

Φ0F=HF:HisinfiniteandhasthepropertyΦ0.E4

F satisfies the selection principle S1Φ0Ψ0 if S1Φ0FΨ0F holds. Identifying the countable sets of functions with sequences of functions, we say that F satisfies the sequence selection principle S1Φ0Ψ0 if S1Φ0FctblΨ0Fctbl holds (compare, e.g., [8]: for each sequence of sequences fn,m:mω:nω of functions from F with the property Φ0, there exists a functions αωω such that fn,αn:nω has the property Ψ0.) Similarly for Sfin and Ufin.

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3. Some results about the relationship of properties of covers and of families of real functions

The first results about covering properties and the properties of the family of continuous real functions were obtained by W. Hurewicz [9]. He proved the following two theorems.

Theorem 1 (W. Hurewicz [9]). If X is a perfectly normal topological space then the following are equivalent.

  1. X is a UfinOctblO

  2. For every sequence fn:nω of continuous real functions the family fnx:nω:xXωR is bounded.

Theorem 2 (W. Hurewicz [9]). If X is a perfectly normal topological space then the following are equivalent.

  1. X is a UfinOctblΓ

  2. For every sequence fn:nω of continuous real functions the family fnx:nω:xXωR is dominating.

Note that the property UfinOctblΓ of a topological space was introduced and investigated by K. Menger [10].

Proofs of both Theorems may be found, e.g., in L. Bukovský and J. Haleš [11].

A topological space X is a γ-space if every open ω-cover of X has a countable γ-subcover.

F. Gerlits and Z. Nagy [12] proved.

Theorem 3 (F. Gerlits and Z. Nagy [12]). If X is a Tychonoff topological space then the following are equivalent:

  1. CpX is Fréchet.

  2. X is a γ-space.

  3. X is an S1ΩΓ-space.

A topological space X has countable strong fan tightness if AnX and xAn¯, nω imply that there exists a sequence xn:nω such that xnAn and xxn:nω¯.

Theorem 4 (M. Sakai [13]). A Tychonoff topological space has the covering property S1ΩΩ if and only if the topological space CpX has countable strong fan tightness.

That was M. Scheepers [2] who began the systematic study of the covering properties GΦΨ for G=S1,Sfin,Ufin, Φ,Ψ=O,Λ,Ω,Γ.

The first use of upper semicontinuous functions in the study of covering properties was.

Theorem 5 (L. Bukovský [1]). A topological space X is an S1ΓΓ-space if and only if USCpX+ satisfies the selection principle S1Γ0Γ0.

Later on we succeeded to prove a general result.

Theorem 6 (L. Bukovský [14]). Assume that Φ is one of the symbols Ω and Γ, and Ψ is one of the symbols O, Ω, Γ. Then for any couple ΦΨ different from ΩO, a topological space X is an S1ΦΨ-space if and only if USCpX+ satisfies the selection principle S1Φ0Ψ0.

Similarly for Sfin and Ufin.

To describe the selection principles of CpX we need different covers of the topological space X. If φ denotes one of the symbols o, ω or γ, then a φ-cover U is shrinkable, if there exists an open φ-cover V such that

VVUVUV¯UV.E5

The family UV:VVU is a φ-cover as well. The family of all open shrinkable φ-covers of X will be denoted by ΦshX.

Extending the result by L. Bukovský and J. Haleš [11] for S1ΓshΓ we obtain.

Theorem 7 (L. Bukovský [14]). Assume that Φ is one of the symbols Ω, Γ and Ψ is one of the symbols O, Ω, Γ. Then for any couple ΦΨ different from ΩO a normal topological space X is an S1ΦshΨ-space if and only if CpX satisfies the selection principle S1Φ0Ψ0.

Similarly for Sfin and Ufin.

The next result is rather a folklore.

Theorem 8. If X is a regular topological space, then for Φ=O,Ω and Ψ=O,Ω,Γ we have Φsh=Φ and therefore

S1ΦshΨS1ΦΨ.E6

Corollary 9. Let X be a normal topological space. Then for Φ=O,Ω and Ψ=O,Ω,Γ the following are equivalent:

  1. CpX satisfies the selection principle S1Φ0Ψ0

  2. The family USCX+ satisfies the sequence selection principle S1Φ0Ψ0.

  3. The family CX satisfies the sequence selection principle S1Φ0Ψ0.

Note that Theorem 4 is a special case of the Corollary.

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4. The measurable covers and functions

Let EPX be as above, i.e. ,XE. A real function fXR is upper E-semimeasurable if for every real aR, the set xX:fx<a belongs to E. A real function fXR is E-measurable if for every reals a<bR, including a=,b=, the set xX:a<fx<b belongs to E. We denote by USMXE the set of all real upper E-semimeasurable functions defined on X. Similarly, we denote by MXE the set of all real E-measurable functions defined on X. Note that if E is a σ-algebra, then MXE=USMXE.

If Xτ is a topological space then MXτ=CX and USMXτ=USCX.

Theorem 10 (L. Bukovský [15]). Assume that Φ is one of the symbols Ω or Γ, Ψ is one of the symbols O, Ω or Γ, and ΦΨΩO. Let E be a family of subsets of a set X, ,XE. If Ψ=Γ, we assume that E is closed under finite intersections.

  1. X possesses the covering property S1ΦEΨE if and only if USMXE+ satisfies the selection principle S1Φ0Ψ0.

  2. Similarly for Sfin.

  3. If E is closed under finite unions, then X possesses the covering property UfinΦEΨE if and only if USMXE+ satisfies the selection principle UfinΦ0Ψ0.

If E is a σ-algebra, then the family USMXE+ may be replaced by MXE.

For E=τ you obtain Theorem 6. For E=BOREL you obtain some results of M. Scheepers and B. Tsaban [16].

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5. Countable covers

A countable family may be considered as a sequence. If U=Un:nω is an (open) cover, then V=inUi:nω is an (open) γ-cover. Some covering properties true for U remain true also for V. E.g., sometimes choosing finite subsets of V is same as choosing finite subsets of U.

If U is uncountable you cannot construct the γ-cover V. Everything you can do is to construct an (open) ω-cover V=W:WUfinite. That is the essence of different behavior of countable and uncountable covers.

W. Just, A. Miller, M. Scheepers and P. Sczeptycki [3] systematically studied the countable covering property GΦctblΨ for G=S1,Sfin,Ufin, Φ,Ψ=O,Λ,Ω,Γ. They obtained several equivalences and as the result the Scheepers Diagram for Countable Covers (Figure 1).

Figure 1.

Sheepers’ diagram for countable covers.

Every countable covering property GΦΨ, where Φ,Ψ are one of the symbols O,Λ,Ω,Γ and G is one of the symbols S1,Sfin,Ufin, is equivalent to some covering property in the Scheepers Diagram for Countable Covers. We do not know whether the thick arrow SfinΓΩUfinΓΩ is reversible. All other arrows of the Diagram are at least consistently not reversible.

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6. Sheepers’ diagram for arbitrary covers

The Sheepers’ Diagram for Countable Covers is valid only for countable covers. If we allow also uncountable covers, some equivalences, used for simplifying the diagram for countable covers, are generally false. We must make the corresponding corrections.

The first simplification consisted in equivalencies of UfinΦΨ for different Φ. Not all of those equivalences are true for uncountable covers. Considering the countable covers, the following properties are equivalent for Ψ=O,Λ,Ω,Γ:

UfinOΨUfinΩΨUfinΓΨ.E7

However, allowing uncountable covers we have only

UfinΩΨUfinOΨE8

for Ψ=O,Λ,Ω,Γ – see Example 11.

In [3] the authors have shown that

S1ΓΓSfinΓΓ.E9

The equivalence

S1ΩΓSfinΩΓE10

remains true also for uncountable covers (an SfinΩΓ-space is a γ-space). One can easily see that for Φ=O,Ω,Γ we have

SfinΩOSfinOO.E11

However, (11) is false at least consistently for SfinΓΦ and UfinΓΦ.

Example 11. Assume that Φ=O,Ω,Γ. Assume that b>1. Then by Theorems 4.6, 4.7 of [3] and (9), the discrete space of cardinality 1 is SfinΓΦ and is not Lindelöf. On the other side, SfinOΦ and UfinOΦ imply that X is Lindelöf.

Example 12. No topological space is SfinOΛ. So neither SfinOΩ nor SfinOΓ. Indeed, if X is an infinite Hausdorff topological space, fix a point aX and an open neighborhood VX of a. For every xV take an open neighborhood Ux of x not containing a. Then U=Ux:xX\VV is an open cover of X and no subcover of U is a λ-cover.

Some covering properties are omitted, since they are equivalent with some others included in the Figure 2. We present those equivalences. Always the former member of an equivalence is included in the Diagram and the latter member is omitted.

Figure 2.

Sheepers’ Diagram for Arbitrary Infinite Covers.

First, take into accunt the equivalences (8)(11). M. Scheepers [2] in Corollaries 5 and 6 has shown that SfinΓΛUfinΓO and S1ΓΛS1ΓO. Evidently [UfinΩΛUfinΛΛUfinOΛ.

Taking in account all mentioned results, we obtain the diagram for arbitrary covers (Figure 2).

We do not know whether the eleven thick arrows of the Scheepers’ Diagram (reversible for countable covers) are reversible for arbitrary covers. The other arrows are not reversible either by Figure 3 of [3] and corresponding results or by Example 11.

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7. Fσ-subsets and covering properties

If U is a family of subsets of X, AX, we set

UA=UA:UU.E12

Let AX. If UA is open in the subspace topology, then there exists an open set UX such that U=AU. Using the Axiom of Choice we choose one such set U for each set UA open in A. If U is an open (in the subspace topology) cover of A then we set U=U:UU. Then U=UA. Note that U need not be a cover of X.

By Theorem 3.1 of [3] we have.

Theorem 13 (W. Just, A. Miller, P. Sczeptycki and M. Scheepers [3]). Let Φ,ΨOΛΩΓ, G being one of S1,Sfin,Ufin. If a topological space X possesses the covering property GΦΨ, FX is closed, then F with the subspace topology possesses the property GΦΨ as well. Moreover, if f:XontoY is continuous, then Y possesses the property GΦΨ as well.

Let S be a “topological” property of topological spaces, EPX, XE. We say that S is hereditary for E if assuming that X has the property S, every AE endowed with the subspace topology has the property S as well. If E=PX then we simply say that S is hereditary on X.

By Theorem 13, for any Φ,ΨOΛΩΓ, G being one of S1,Sfin,Ufin, the covering property GΦΨ is hereditary for closed subsets.

In [8], M. Scheepers proved that addS1ΓΓh. Since h>0, by Theorem 13 we obtain.

Corollary 14. The covering property S1ΓΓ is hereditary for Fσ-subsets of X.

We prove the following result.

Theorem 15. Let ΦOΩΓ and ΨOΛΩ. Let G be one of S1,Sfin,Ufin. The covering property GΦΨ is hereditary for Fσ-subsets of X.

Proof: Let F=nFn, where for each n the set Fn is closed and FnFn+1. Let the countable sequence of open φ-covers of F be bijectively enumerated as

Un,m:nmω.E13

Then Un,mFn is a φ-cover of Fn. Apply S1ΦΨ to the sequence Un,mFn:mω for every n. We obtain sequences Vn,mUn,mFn:mω such that every family Vn,m:mω is a ψ-cover of Fn. Let Un,mUn,m be such that Vn,m=Un,mFn.

One can easily see that the family Un,m:nmω is a ψ-cover of F. Since Un,mUn,m for every n and m, we obtain that F possesses the covering property S1ΦΨ.

For G=Sfin,Ufin the proof is similar.□

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8. σ-space

A topological space X is said to be a σ-space, if every Gδ-subset of X is an Fσ-set. Consequently, every Borel subset of a σ-space is an Fσ-set.

J. Haleš [4] proved.

Theorem 16. Let X be a perfectly normal topological S1ΓΓ-space. The covering property S1ΓΓ is hereditary on X if and only if X is a σ-space.

We obtain an easy Corollary.

Corollary 17. Let Φ be one of the symbols O,Λ,Ω,Γ. Let G be one of S1,Sfin. Assume that X is a perfectly normal topological space. If X possesses the covering property GΦΓ and GΦΓ is hereditary on X, then X is a σ-space.

Proof: Note that

GΦΓGΓΓS1ΓΓ,E14

(if G=Sfin use (9)) and ΓΦ. □

Following Haleš’ proof of Theorem 16 we obtain.

Theorem 18. Let X be a σ-space, Φ being one of the symbols O,Λctbl,Γ and Ψ being one of O,Λ,Ω,Γ. Let G be one of S1,Sfin,Ufin. Then GΦΨ is hereditary on X.

Proof: Assume that X is a topological GΦΨ-space, where Φ=O,Λctbl,Γ and Ψ=O,Λ,Ω,Γ. Assume also that X is a σ-space and AX. Let Un:nω be a sequence of open (in the subspace topology) φ-covers of A.

If Φ=O then UnB is an o-cover of B=nUnA. Since B is a Borel set, therefore an Fσ-set in X, by Theorem 15, the set B possesses the covering property GOΨ. So, for each nω there exists a UnUnB or finite set WnUnB such that Un:nω, or nWn, or Wn:nω is a ψ-cover of B, respectively. Then Un:nωA, or nWnA, or Wn:nωA is an open ψ-cover of A, respectively.

Let Φ=Λctbl. Since each Un is countable, we can assume that Un,m:mω is a bijective enumeration of Un. The family Un,m=Un,k:km is a λ-cover of A. If we set B=n,mUn,mA then each UnB is a λ-cover of B. Continue as above.

If Φ=Γ we can assume that each Un is countable. For every nω let Un,m:mω be a bijective enumeration of Un. Then UnB is a γ-cover of B=nkmkUn,mA. The set B is Borel, therefore Fσ. By Theorem 15, B is an GΓΨ-space. Continue as above.□

For Φ=Ψ=Γ, G=S1, we obtain one implication of the Haleš’ Theorem 16.

Corollary 19. Let Φ be one of the symbols O,Λctbl,Γ. Let G be one of S1,Sfin. The covering property GΦΓ is hereditary on a perfectly normal topological space X if and only if X is a σ-space.

Corollary 20. Let Φ be one of the symbols O,Λctbl,Γ. Let G be one of S1,Sfin. Assume that a perfectly normal topological space X possesses the covering property GΦΓ. Then X is hereditary GΦΓ if and only if X is hereditary S1ΓΓ.

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9. Remarks

I have obtained a short time for writing this paper. So, I have no time to collect all results known before proving Theorems 5, 6, 7, 10, 15 and 18. For a partial presentation of such known results see [14, 15].

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Classification

2010 MSC: 54C35, 54C20, 54D55

References

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Notes

  • If W is finite then minW=h, where hx=minfx:f∈W for every x∈X. min∅=1.
  • A function f∈XR is upper semicontinuous if the set x∈X:fx<a is open for every real a∈R.

Written By

Lev Bukovský

Reviewed: 21 September 2021 Published: 24 June 2022