REVIEW 6 major objections 7 minor 23 references
Cardinal functions and mappings associated with the space of quasi-continuous functions equipped with topology of point-wise convergence
T0 review · 6 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The space of quasi-continuous functions mirrors the domain's cardinal size.
desk verdict The paper's main cardinal inequalities and mapping theorems don't hold as stated; two are false, and the proofs of the others misuse a separation lemma that cannot prescribe values on whole open sets. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Carrying the argument is a family of quasi-continuous indicator functions: functions that take one constant value on a prescribed open set and another constant value on its complement, built from a separation lemma for regular spaces (Lemma 2, taken from [19]). These indicator functions let the proofs code points, open covers, and candidate networks of $X$ inside $Q_P(X)$ or $Q_P(X,Y)$, so a lower bound on a cardinal invariant of the function space yields a lower bound on the corresponding covering or network invariant of $X$. For the mapping results, the topology of point-wise convergence makes point evaluations continuous, which is what the paper uses to transfer surjectivity and denseness statements back to the domain.
What would settle it
Take $X=\mathbb{R}$, $V=(0,1)$, and define $f(x)=1$ for $x\in V$ and $f(x)=0$ otherwise. At $x=0$, every non-empty open subset of a neighbourhood of $0$ contains points of $V$ where $f(x)=1$, so $f$ is not quasi-continuous at $0$; checking this single case directly would falsify the assumption used in the proof of Theorem 5. A second concrete check is to test the claimed surjectivity of the restriction map for $X=\mathbb{R}$ and $Y=(0,1)$ by asking whether every quasi-continuous function on the open interval extends to a quasi-continuous function on all of $\mathbb{R}$ constant outside $Y$.
Extended reading notes
Core claim
The central claim, stated on the paper's own terms, is that for an ordered Hausdorff space $X$ one has $nw(Q_P(X))=w(Q_P(X))=|X|$, and for a regular space $X$ with a metric space $Y$ one has $wc(X)\le \psi(Q_P(X,Y))$, while $nw(X)\le \psi(Q_P(X,Y))$ for any target space $Y$. The paper further derives $wc(X)\cdot\log(nw(X))\le iw(Q_P(X))$ for regular $X$. On the mapping side, it claims that the restriction map $\pi_Y:Q_P(X)\to Q_P(Y)$ is open, continuous, and surjective whenever $Y$ is an open subspace of $X$, and that for every continuous surjection $r:Y\to Z$ the induced map $r_*:Q_P(X,Y)\to Q_P(X,Z)$ has image dense in $Q_P(X,Z)$. Along the way it gives corollaries linking countable tightness of $Q_P(X)$ to the Lindelöf property of $X$, and countable pseudocharacter of a compact subspace of $Q_P(X,Y)$ to separability of $X$.
Load-bearing premise
The main inequalities rely on being able to construct quasi-continuous functions that are constant (say $0$) on an entire prescribed open set and constant (say $1$) on its complement, but the cited separation lemma only guarantees prescribed values at a point and on a closed set, and a function that is constant on both sides of a boundary can fail the quasi-continuity condition exactly on that boundary.
Editorial extensions
If this is right
- For any ordered Hausdorff space $X$, the pointwise convergence space $Q_P(X)$ has weight and network weight exactly $|X|$, so its size in the sense of bases and networks coincides with the size of the domain.
- For regular $X$ and metric $Y$, every family of open neighbourhoods of the constant function that separates it from the rest of $Q_P(X,Y)$ is at least as large as a weak cover of $X$, so $wc(X)\le \psi(Q_P(X,Y))$.
- If the tightness of $Q_P(X)$ is countable, then every open cover of a regular space $X$ has a countable subcover; that is, $X$ is Lindelöf.
- If some compact subspace of $Q_P(X,Y)$ has countable pseudocharacter, then $X$ is separable.
- When $Y$ is an open subspace of $X$, the restriction map $Q_P(X)\to Q_P(Y)$ is claimed to be open, continuous and onto; and for a continuous surjection $r:Y\to Z$, the image of $Q_P(X,Y)$ under $r_*$ is dense in $Q_P(X,Z)$.
Reading between the lines
- If the claimed equalities hold, $Q_P(X)$ behaves like $C_p(X)$ for weight-type invariants on ordered Hausdorff domains, so other classical $C_p$-theoretic inequalities may have quasi-continuous analogues.
- The proof template suggests a broader recipe: any space $X$ admitting enough quasi-continuous indicator functions that separate points from open sets would satisfy the same lower bounds, and orderedness may not be essential.
- A testable extension is whether the induced map $r_*$ is not merely dense-valued but surjective when $Y$ is compact and $Z$ is Hausdorff; if surjectivity fails, the denseness claim is the best possible in that setting.
- The extension step for restriction maps to open subspaces is the fragile point; testing it on $X=\mathbb{R}$ and $Y=(0,1)$ would show how far the surjectivity claim can be pushed.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies cardinal invariants (network weight, Lindelöf degree, tightness, weak covering number, pseudocharacter, i-weight) of the spaces Q_P(X) and Q_P(X,Y) of quasi-continuous functions with the topology of pointwise convergence, and it also examines restriction and induced maps on these spaces. The main claims are: for an ordered Hausdorff space X, nw(Q_P(X))=w(Q_P(X))=|X| (Theorem 1); for regular X, L(X)≤t(Q_P(X)) (Theorem 2) and wc(X)≤ψ(Q_P(X,Y)) for metric Y (Theorem 3); nw(X)≤ψ(Q_P(X,Y)) for regular X (Theorem 5); wc(X)·log(nw(X))≤iw(Q_P(X)) (Theorem 6); the restriction map to an open subspace is open, continuous, and surjective (Theorem 8); and the induced map has dense image (Theorem 11). The paper also contains a proof of continuity of the pointwise product map (Theorem 10).
Significance. If correct, the cardinal inequalities would extend classical C_p-theory results to quasi-continuous function spaces and would be of interest to researchers working on generalized continuity and function spaces. The paper also collects relevant background from [19] and [21], and Theorem 10 contains a correct elementary epsilon-delta proof for the continuity of the product map. However, the central results are not established: the proofs repeatedly use characteristic functions of arbitrary open sets as if they were quasi-continuous, and they apply the quoted separation lemma outside its hypotheses. At least Theorems 2, 3, 5, 8, and 11 have load-bearing gaps, and the extension step in Theorem 8 is demonstrably false in simple cases. The paper therefore cannot be accepted in its present form.
major comments (6)
- [Section 3, Theorems 1, 2, and 5] The proofs rely on the assertion that the characteristic function of an arbitrary open set is quasi-continuous. This is false. If O is open and x∈∂O, then for f=1_O and f(x)=1, every nonempty open G⊂U meets X\O where f=0, so f(G) is not contained in (1/2,2); the case f(x)=0 is symmetric. Lemma 2 ([19, Lemma 4.12]) prescribes values only at one point and on a closed set, so it cannot produce a function with f_A(O_A)={0} and f_A(X\O_A)={1}. The same defect invalidates the functions f_r in Theorem 1, and the injection argument there is also incorrect: for r1<r2, both f_{r1} and f_{r2} take value 0 at r2, so f_{r1}∈W(f_{r2},{r2},ε). Thus Theorems 1, 2, and 5 are not established as written.
- [Section 3, Theorem 2] The proof asserts 'zero function f0∈P'. For A∈F′ with O_A a proper nonempty open set, the constructed f_A equals 0 on O_A and 1 on X\O_A, so f_A is not the zero function. Consequently the tightness step 'there exists P′⊂P with f0∈P′' is unjustified, and the conclusion L(X)≤t(Q_P(X)) does not follow.
- [Section 3, Theorems 3 and 4] In Theorem 3, Lemma 2 is applied with E=⋃J, but ⋃J is a union of open sets and need not be closed, whereas Lemma 2 requires a closed E. The same problem occurs in Theorem 4, where the set D is countable but not proved closed. Theorem 4 also uses the expression |f(x)|<b′ for an arbitrary ordered space Y without defining an absolute value or a norm; such an inequality is not meaningful in a general ordered topological space. The inequalities wc(X)≤ψ(Q_P(X,Y)) and the separability criterion are therefore unsupported.
- [Section 4, Theorem 8] The extension h defined by h(x)=g(x) for x∈Y and h(x)=1 for x∈X\Y is not quasi-continuous in general. For X=ℝ and Y=(0,1) with g≡0, the function h equals 0 on (0,1) and 1 elsewhere; at x=1, for the neighbourhood V=(1/2,2) of h(1), every nonempty open G⊂U meets (0,1) where h=0, so h(G) is not contained in V. Hence π_Y(Q_P(X))=Q_P(Y) is not proved. The openness proof is also defective: the sets V_i appearing in the second half of the proof are never defined, and the equality π_Y(W(f,{x1,...,xk},ε))=W(π_Y(f),{x1,...,xl},ε) is not established.
- [Section 4, Theorem 11] The proof assumes that for arbitrary finite {x_i} and nonempty open sets U_i⊂Y there exists f∈Q_P(X,Y) with f(x_i)∈U_i, so that the basic open set [x1,...,xn;U1,...,Un] is nonempty. This is not automatic. For example, if X is connected and Y is the two-point discrete space, every quasi-continuous f:X→Y is constant, so a basic open set prescribing f(x1)=0 and f(x2)=1 is empty. The argument only shows that any f in such a set maps into the target subbasic open set; it does not show that one exists. The density of r_*(Q_P(X,Y)) in Q_P(X,Z) is therefore not established.
- [Section 3, Theorem 6] The proof rests on the inequality ψ(Z)·log(nw(Z))≤iw(Z) for arbitrary Z, stated without proof or reference. This is not a standard inequality; known results give iw(Z)≤ψ(Z)·log(nw(Z)) in the opposite direction, so the claimed inequality is suspect. Since this is the only justification for the product inequality, Theorem 6 is unsupported.
minor comments (7)
- [Section 2, Preliminaries] The sentence on cardinal numbers reads 'The first infinite (countable) cardinal no., uncountable (second uncountable) cardinal no., and arbitrary cardinal number are denoted by ℵ0, ℵ0, and η'; the second symbol should be ℵ1, not ℵ0.
- [Section 2, displays (21) and (22)] The symbol ω(Q_P(X,Y)) is used without definition; elsewhere the weight is denoted w(X). The notation should be unified.
- [Section 3, first paragraph] The word 'matrizable' should be 'metrizable'.
- [Section 4, Theorem 8 proof] The phrase 'It is oblivious π_Y(Q_P(X)) ⊂ Q_P(Y)' should read 'It is obvious that'.
- [Section 4, paragraph after Theorem 8] The statement that semi-continuity and quasi-continuity are equivalent for single-valued functions is not correct in general; semi-continuity does not imply quasi-continuity. The authors should state the precise result they intend.
- [Section 4, Theorem 9] Theorem 9 is stated without proof or explicit citation; a short proof or a reference should be supplied.
- [Section 5, Conclusion] The sentence 'we found that a regular space is regular whenever the pseudocharacter of a compact subset of space Q_P(X,Y) is countable' appears to say 'regular' twice; the intended conclusion is presumably that X is separable.
Circularity Check
No circularity: the paper relies on external lemmas and known results; its proof gaps are correctness issues, not self-referential reasoning.
full rationale
The paper's derivation chain is not circular. Its central theorems invoke external lemmas from Kumar–Tyagi [19] and Hola–Holy [21], then apply those lemmas to construct quasi-continuous functions. The conclusions of the theorems are not used as inputs to themselves, and no fitted parameter is renamed as a prediction. The proofs do contain serious verification gaps: Theorem 2 and Theorem 5 assume that characteristic functions of arbitrary open sets are quasi-continuous and that Lemma 2 can prescribe values on an entire open set, while Theorem 8 and Theorem 11 contain unjustified extension and density arguments. However, these are failures of lemma application or proof validity, not circularity. The only self-citations ([6], [14], [15]) appear in the introduction as background and are not load-bearing. Therefore the paper scores 0 for circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption Lemma 1: a function that is locally constant on a neighborhood of each point is quasi-continuous (cited from [21, Lemma 4.2]).
- domain assumption Lemma 2: in a regular space X, for a point x not in closed set E and y1,y2 in Y, there is f in Q_P(X,Y) with f(x)=y1 and f(E)={y2} (cited from [19, Lemma 4.12]).
- domain assumption Lemma 3: the evaluation map e_x:Q_P(X,Y)→Y is continuous (cited from [19, Lemma 5.5]).
- domain assumption For any topological space Z, ψ(Z)·log(nw(Z)) ≤ iw(Z).
Cite this review
Pith. "Pith review of Cardinal functions and mappings associated with the space of quasi-continuous functions equipped with topology of point-wise convergence." pith.science (2026). https://pith.science/paper/2VKXKHZK
@misc{pith2026241202188,
author = {Pith},
title = {Pith review of: Cardinal functions and mappings associated with the space of quasi-continuous functions equipped with topology of point-wise convergence},
year = {2026},
howpublished = {\url{https://pith.science/paper/2VKXKHZK}},
note = {Machine review of arXiv:2412.02188}
}
abstract
Cardinal functions provide valuable insight into the topological properties of spaces, helping to analyze and compare spaces in terms of their covering, convergence and separation properties. This paper focuses on investigating cardinal functions like network weight, Lindel\"of degree, tightness, weak covering, pseudocharacter, and $i$-weight, for the spaces $Q_{P}(X)$ and $Q_{P}(X,Y)$ of quasi-continuous functions under the topology of point-wise convergence. In addition to these, we also investigate properties of restriction and induced maps associated with the spaces $Q_{P}(X)$ and $Q_{P}(X,Y)$.
Reference graph
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Reviewed August 11, 2026 · model on record in the stance chip above.
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