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REVIEW 5 major objections 4 minor 23 references

Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets

T0 review · 5 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For Hausdorff X, tightness of QC(X) equals the compact Lindelöf number of X.

desk verdict The paper's intended theorems are plausible analogues of known C_c(X) results, but the proofs rely on cutoff functions that are not quasicontinuous and on false compactness assumptions, so the central claims are not established. read the letter →

arxiv 2412.02209 v1 pith:NQQMTOZK submitted 2024-12-03 math.GN

Chander Mohan Bishnoi , Sanjay Mishra This is my paper · ORCID
classification math.GN MSC 54C3554A2554C0854C3054D10
keywords quasicontinuousfunctionstopologyofuniformconvergenceoncompactatightnessdensityfanFrechet-Urysohnpropertyk-coverscompactLindelöfnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the space $QC(X)$ of real-valued quasicontinuous functions on a topological space $X$, equipped with the topology of uniform convergence on compact subsets. Its central aim is to show that cardinal invariants of this function space are controlled by covering properties of $X$: the tightness of $QC(X)$ equals the compact Lindelöf number $kL(X)$ for Hausdorff $X$, and density tightness, fan tightness, and the Frechet-Urysohn property are characterized in similar terms. If true, these results turn questions about how hard it is to approximate one quasicontinuous function by others into concrete questions about how compact subsets of $X$ can be covered by open sets. A reader who wants to know whether a quasicontinuous function space behaves discretely or sequentially can now read the answer directly from the size of $X$'s compact-cover families.

What carries the argument

The central object is the function space $QC(X)$: all real-valued quasicontinuous functions on a topological space $X$, topologized by uniform convergence on compact subsets. The argument is carried by $k$-covers — families of open sets such that every compact subset of $X$ lies inside some member — and by the compact Lindelöf number $kL(X)$, the least cardinal of a subcover that still covers every compact set. The bridge between the two is a construction that assigns to each member $U$ of a $k$-cover a quasicontinuous function taking value $0$ on $U$ and $1$ outside; closure properties of families of such functions in $QC(X)$ are then translated back into covering properties of $X$. A second structural tool is homogeneity (Lemma 3.8): translation by a continuous function is a homeomorphism of $QC(X)$, so conclusions obtained at the zero function extend to every function.

What would settle it

Take $X = \mathbb{R}$, let $U = (0,1)$, and define $f$ by $f(x)=0$ on $U$ and $f(x)=1$ on $\mathbb{R}\setminus U$. At either boundary point, every neighbourhood contains points where $f$ takes both values, so $f$ fails the definition of quasicontinuity; checking this function against the definition settles whether the cutoff construction used in the proof of Theorem 3.4 is valid.

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Extended reading notes

Core claim

The paper's central claim is a structural identity: for any Hausdorff space $X$, the tightness of $QC(X)$ is exactly the compact Lindelöf number $kL(X)$ (Theorem 3.4). Tightness here measures the smallest cardinal $\kappa$ such that, whenever a point lies in the closure of a set, it lies in the closure of a subset of size at most $\kappa$; the compact Lindelöf number measures the smallest cardinal $\lambda$ such that every open cover of the compact subsets of $X$ contains a subcover of cardinality at most $\lambda$. The proof proceeds through $k$-covers and cutoff functions built from them, and a companion result (Theorem 4.2) states that for locally compact Hausdorff $X$, countable tightness of $QC(X)$, the Frechet-Urysohn property of $QC(X)$, and $\sigma$-compactness of $X$ are equivalent.

Load-bearing premise

The arguments rely on treating functions that are constant on an open set and constant (with a different value) on the rest of the space as quasicontinuous, and on treating finite closed sets in regular spaces as compact.

Editorial extensions

If this is right

  • If $X$ is a second countable Hausdorff space, then $QC(X)$ has countable tightness (Corollary 3.5).
  • For a locally compact Hausdorff space $X$, $QC(X)$ is Frechet-Urysohn if and only if $X$ is $\sigma$-compact (Theorem 4.2).
  • For a Hausdorff space $X$, the density tightness of $QC(X)$ equals its tightness (Theorem 3.7).
  • If $QC(X)$ is Frechet-Urysohn for a Hausdorff space $X$, then every open $k$-cover of $X$ has a countable $k$-subcover (Theorem 4.1).
  • For a locally compact metric space $X$, $QC(X)$ is Frechet-Urysohn if and only if $X$ is separable (Corollary 4.3).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the equality $t(QC(X)) = kL(X)$ holds, it suggests a broad dictionary between cardinal invariants of generalized-continuous function spaces and covering numbers of the base space; the same $k$-cover technique might be adapted to other classes of functions that admit a similar locally-constant cutoff construction.
  • The cutoff-function step is the natural stress point: a single Hausdorff space where such a piecewise-constant function fails to be quasicontinuous would not refute the equality outright but would force a new proof, and such examples likely exist whenever members of the $k$-cover have nonempty boundary.
  • A testable extension: replace 'quasicontinuous' with 'Baire-one' or 'Darboux' functions and compare the resulting tightness with $kL(X)$; divergence would show which properties of quasicontinuity are actually carrying the cardinal bound.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The manuscript studies cardinal invariants of the space QC(X) of real-valued quasicontinuous functions on a topological space X, equipped with the topology of uniform convergence on compact subsets. The main claimed results are: Theorem 3.4, t(QC(X)) = kL(X) for Hausdorff X; Theorem 3.7, dt(QC(X)) = t(QC(X)); Theorems 3.9 and 3.10, characterizations of fan tightness and strong fan tightness in terms of open k-covers; Theorem 3.12, d(QC(X)) ≤ kcof(X) for regular X; and in Section 4, equivalences between countable tightness, the Frechet-Urysohn property of QC(X), and sigma-compactness of a locally compact Hausdorff space X, together with results on kf-covers and Whyburn spaces. The paper consists mostly of proofs that use cutoff functions built from open sets U and asserts their quasicontinuity by Lemma 2.4.

Significance. The topic is natural and the proposed analogues of classical Cp-theory theorems, such as the Ferrando-Moll theorem for C(X), are worth investigating. If the claims were correct, the equality t(QC(X)) = kL(X) and the locally compact equivalence between the Frechet-Urysohn property, countable tightness, and sigma-compactness would be useful additions to the theory of quasicontinuous function spaces. However, the manuscript contains no machine-checked proofs or reproducible artifacts, and the central arguments are invalid as written: the cutoff functions used throughout Section 3 are generally not quasicontinuous, and several topological assertions (closed sets in regular spaces are compact; open sets in locally compact spaces are compact) are false. The main results are therefore not supported by the present proofs.

major comments (5)
  1. [Theorem 3.4] The functions f_K defined by f_K(U_K) = {0} and f_K(X\U_K) subset {1} are not shown to be quasicontinuous. Lemma 2.4 applies only to locally constant functions, and such a cutoff is not locally constant at boundary points of U_K; it need not be quasicontinuous at all. For example, with X = R and U = R\{0}, the function f(0)=1 and f(x)=0 for x≠0 fails quasicontinuity at 0: for the neighbourhood (-1,1) of 0 and the open set (1/2,3/2) around f(0), every nonempty open W subset (-1,1) contains a point x≠0 with f(x)=0 not in (1/2,3/2). The same problem occurs in the second half of the proof, where f_U_K is defined to equal f on U_K and 1 outside U_K. Since this construction is used in both directions of Theorem 3.4 and in Corollaries 3.5, 3.6, and Theorem 4.1, the central equality t(QC(X)) = kL(X) is not established.
  2. [Lemma 3.11] The proof of Lemma 3.11 begins with the assertion that F1, F2, ..., Fn are compact subsets of X; in a regular space, closed sets need not be compact. The lemma is false as stated (for example, in R with the usual topology, F = [0,∞) is closed but not compact), so Theorem 3.12, which invokes the lemma to construct the family D, is not proved. Moreover, the function f defined by f(x) = y_i f_i(x) if x in F_i and f(x)=0 otherwise is not shown to be quasicontinuous by the cited lemma.
  3. [Theorem 4.2] In the proof of (1)=>(3) of Theorem 4.2, the authors take a countable k-subcover U' of an open k-cover U and set M = {U : U in U'}; they then assert that 'Since X is locally compact, each set in M is compact.' Open sets in a locally compact space are not generally compact. Consequently the conclusion that M is a cofinal family in K(X) and that X is hemicompact or sigma-compact does not follow. The claimed equivalence between countable tightness of QC(X), the Frechet-Urysohn property, and sigma-compactness of X is therefore unsupported.
  4. [Theorem 4.6] The functions f_n^m in the proof of Theorem 4.6 are required to satisfy f_n^m|F_n^m ≡ 1/n and f_n^m|(X\U_n^m) ≡ 1. Since the hypotheses only give F_n^m subset U_{m+1}^n, and not F_n^m subset U_m^n, the two defining conditions can assign conflicting values at points of F_n^m ∩ (X\U_n^m). Even when the sets are disjoint, the resulting cutoff is not automatically quasicontinuous by Lemma 2.4. The subsequent assertions that the zero function h lies in S and that the Whyburn property yields the set F are also not justified. Thus Theorem 4.6 is not established.
  5. [Theorems 3.9 and 3.10] The proofs of Theorems 3.9 and 3.10 repeatedly define auxiliary functions by h(x)=g(x) for x in U and h(X\U) subset {0}, and then assert h in QC(X) by Lemma 2.4; as in Theorem 3.4, this cutoff is generally not quasicontinuous at boundary points of U. In addition, the statements 'g_m(X) = (-1/m, 1/m)' and 'f_m(X) = (-1/m, 1/m)' are not meaningful for real-valued functions, since an image is a subset and equality to an interval would force the function to be constant on X with that image. This makes the convergence arguments in the two cases unverifiable. Hence the fan-tightness and strong-fan-tightness characterizations are not supported.
minor comments (4)
  1. [Theorem 3.7] The notation 't(U_X)' should be 't(QC(X))', and 'Theorem (2)' should refer to the inequality dt(Z) ≤ t(Z) from the preliminaries rather than an unnamed theorem.
  2. [Throughout] The manuscript uses 'X/U' for set difference, which is confusing; the standard notation is X\U. The abstract and proof text also contain many typos, such as 'Frechet-Uryshon', 'Thud', and 'X/A' in the introduction.
  3. [Corollary 4.3] In the proof of (2)=>(3), the statement that sigma-compactness of X implies separability of QC(X) needs an explicit argument; it is not an immediate consequence of sigma-compactness and is not otherwise proved in the paper.
  4. [Theorem 4.5] The proof refers to 'f1' when defining the neighborhood W(f1, K_i, epsilon); the intended function is presumably the constant function g1. The notation should be made consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's claims are not derived from their own conclusions or from load-bearing self-citations; the serious defects are invalid proof steps, not circular reasoning.

full rationale

The derivation chain in this paper does not exhibit the characteristic reductions required for circularity. The central theorem, t(QC(X)) = kL(X), is proved by constructing cutoff functions that take value 0 on an open set U and value 1 on X\U, then invoking Lemma 2.4. This application is mathematically invalid, because such a cutoff is not locally constant at boundary points and need not be quasicontinuous; indeed, the proof of Theorem 3.4, Theorem 4.1, and related results all inherit this defect. However, this is a correctness problem, not a circularity problem: the theorem is not being assumed, fitted, or defined in terms of itself. Lemma 3.11's false assertion that finite disjoint closed subsets of a regular space are compact is likewise an invalid step, not a circular one. The paper's references to its own authors' prior work are not load-bearing for the main derivations, and the central claims are not justified solely by self-citation. No parameter is fitted to data and then renamed a prediction, and no result is equivalent to its input by construction. Therefore the appropriate circularity score is 0, while the mathematical validity of the proofs is a separate concern.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on standard ZFC topology, the definition of quasicontinuity from Neubrunn, and cited lemmas from Hola-Holy and Kumar-Tyagi. Additionally it silently uses two false assumptions: that closed sets in regular spaces are compact, and that open sets in locally compact spaces are compact.

assumptions (6)
  • standard math ZFC set theory and standard topological definitions
    The paper works in standard set-theoretic topology; no nonstandard axioms are introduced.
  • domain assumption Definition of quasicontinuity taken from Neubrunn
    The paper relies on the standard definition of quasicontinuous functions as given in the introduction.
  • standard math Lemma 2.4: every locally constant real-valued function on a Hausdorff space is quasicontinuous
    Cited from Hola-Holy; the paper applies this lemma to cutoff functions that are not locally constant, exceeding its scope.
  • domain assumption Lemma 2.5: quasicontinuous separation between a point and a closed set
    Cited from Kumar-Tyagi; used in Theorem 4.5 to separate two closed sets, which is a stronger property than the lemma provides.
  • ad hoc to paper Closed subsets of a regular space are compact
    Assumed in Lemma 3.11 without proof; false in general regular spaces.
  • ad hoc to paper Open subsets of a locally compact space are compact
    Assumed in Theorem 4.2, part (1) implies (3); false in general locally compact spaces.

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Pith. "Pith review of Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets." pith.science (2026). https://pith.science/paper/NQQMTOZK

@misc{pith2026241202209,
  author       = {Pith},
  title        = {Pith review of: Cardinal Properties of the Space of Quasicontinuous Functions under Topology of Uniform Convergence on Compact Subsets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQQMTOZK}},
  note         = {Machine review of arXiv:2412.02209}
}
abstract

In this paper, we investigate various cardinal properties of the space $Q_{C}X$ of all real-valued quasicontinuous functions on the topological space $X$, under the topology of uniform convergence on compact subsets. It begins by examining the relationship between tightness and other properties in the context of the space $X$, highlighting results such as the alignment of tightness $Q_{C}X$ with the compact Lindel\"of number of $X$ under Hausdorff conditions and the countable tightness of $Q_{C}X$ when $X$ is second countable. Further investigations reveal conditions for the tightness of $Q_{C}X$ relative to $k$-covers of $X$, as well as connections between density tightness, fan tightness, and other properties in Hausdorff spaces. Additionally, we discuss the implications of the Frechet-Urysohn property $Q_{C}X$ for open $k$-covers in Hausdorff spaces. We explore relationships between $Q_{C}X$'s tightness, the Frechet-Urysohn property, and the $\sigma$-compactness of locally compact Hausdorff spaces $X$. Furthermore, we examine the $k_{f}$-covering property and the existence of $k$-covers in the context of Whyburn spaces.

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Aaliya and S

    M. Aaliya and S. Mishra, Space of homeomorphisms under regular topology , Commun. Korean Math. Soc. 38(2023), 1299-1307

    work page 2023

  2. [2]

    Aaliya and S

    M. Aaliya and S. Mishra, Compactness and cardinality of the space of continuous func tions under regular topology, Palest. J. Math. 13(1)(2024), 109-117. 10 CHANDER MOHAN BISHNOI* AND SANJAY MISHRA

    work page 2024

  3. [3]

    Aaliya and S

    M. Aaliya and S. Mishra, Some properties of regular topology on C (X, Y) , Ital. J. Pure Appl. Math. 50(2023), 27-43

    work page 2023

  4. [4]

    Arkhangel’skii, Topological function spaces, Kluwer Academic Publishers(1992), 78

    A.V. Arkhangel’skii, Topological function spaces, Kluwer Academic Publishers(1992), 78

    work page 1992

  5. [5]

    Baire, Sur les functions des varaibles reells , Ann

    R. Baire, Sur les functions des varaibles reells , Ann. Mat. Pura Appl. 3(1899), 1-122

    work page

  6. [6]

    Bishnoi and S

    C.M. Bishnoi and S. Mishra, Quasicontinuous function on strong forms of connected spac e, J. Indones. Math. Soc. 29(1) (2023), 106-115

    work page 2023

  7. [7]

    Engelking, General topology, Heldermann Berlin.(1989)

    R. Engelking, General topology, Heldermann Berlin.(1989)

    work page 1989

  8. [8]

    J. C. Ferrando and S. Moll, Cc(X) Spaces with X Locally Compact , Acta Math. Sin. Engl. Ser. 23(9)(2007), 1593-1600

    work page 2007

Show all 23 references
  1. [9]

    Hola and D

    L. Hola and D. Holy, Quasicontinuous functions and compactness , Mediterr. J. Math. 14(6)(2017), 1-11

    2017

  2. [10]

    Hola and D

    L. Hola and D. Holy, Quasicontinuous subcontinuous functions and compactness , Mediterr. J. Math.13(2016), 4509-4518

    2016

  3. [11]

    Hola and D

    L. Hola and D. Holy, Metrizability of the space of quasicontinuous functions , Topol. Appl. 246(2018), 137- 143

    2018

  4. [12]

    Hola and D

    L. Hola and D. Holy, Quasicontinuous functions and the topology of pointwise co nvergence, Topol. Appl.282(2020), 107301

    2020

  5. [13]

    Hola and D

    L. Hola and D. Holy, Quasicontinuous Functions and the Topology of Uniform Conv ergence on Compacta , FILOMAT.35(2021), 911-917

    2021

  6. [14]

    L. Hola, D. Holy, and W. Moors, USCO and Quasicontinuous mappings , De Gruyter. 81(2021), 107301

    2021

  7. [15]

    Kempisty, Sur les fonctions quasicontinues , Fundam

    S. Kempisty, Sur les fonctions quasicontinues , Fundam. Math. 1(1932), 184-197

    1932

  8. [16]

    Kumar and B.K

    M. Kumar and B.K. Tyagi, Cardinal invariants and special maps of quasicontinuous fu nctions with the topology of pointwise convergence , Appl. Gen. Topol. 23(2)(2022), 303-314

    2022

  9. [17]

    Mishra and A

    S. Mishra and A. Bhaumik, Properties of function space under Cauchy convergence topo logy, Topol. Appl. 338(2023), 108653

    2023

  10. [18]

    T.Neubrunn, Quasi-continuity,Real Anal. Exch. 14(1988), 259-306

    1988

  11. [19]

    Okunev and V.V

    O.G. Okunev and V.V. Tkachuk, Density properties and points of uncountable order for fami lies of open sets in function spaces , Topol. Appl. 122(2002), 397-406

    2002

  12. [20]

    Osipov, Fr´ echet-Urysohn property of quasicontinuous functions, Rocky Mt

    A. Osipov, Fr´ echet-Urysohn property of quasicontinuous functions, Rocky Mt. J. Math. (2023)

    2023

  13. [21]

    Tkachuk, A Cp-theory problem book:Topological and function spaces , Springer,(2010)

    V.V. Tkachuk, A Cp-theory problem book:Topological and function spaces , Springer,(2010)

    2010

  14. [22]

    Tkachuk, A Cp-theory problem book:Special features of function spaces , Springer,(2014)

    V.V. Tkachuk, A Cp-theory problem book:Special features of function spaces , Springer,(2014)

    2014

  15. [23]

    Tkachuk, A Cp-theory problem book:Compactness in function spaces , Springer,(2015)

    V.V. Tkachuk, A Cp-theory problem book:Compactness in function spaces , Springer,(2015). Chander Mohan Bishnoi, Department of Mathematics, Lovely P rofessional University, Punjab, India Email address : chandermohan.cm.b@gmail.com Sanjay Mishra, Department of Mathematics, Amity...

    2015

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