{"id":"8aa133b1-2d38-4403-926d-dab9e8ad81eb","arxiv_id":"2412.02188","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Several claimed theorems about cardinal invariants of quasi-continuous function spaces are invalid: the restriction map need not be surjective and the induced-map image need not be dense.","lead":"Topologists measure the 'size' of spaces made from quasi-continuous functions, a flexible generalization of continuous functions. This paper claims new bounds and mapping properties for those function spaces, but several key proofs are invalid and two main theorems are false.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proofs of Theorems 2 and 5 assume characteristic functions of arbitrary open sets are quasi-continuous; this fails at boundary points, leaving L(X)≤t(QP(X)) and nw(X)≤ψ(QP(X,Y)) unsupported.","rationale":"The reader's verdict is well supported. The weakest point in the cardinal-invariant chain is the assertion that a function which is constant on an arbitrary open set and constant on its complement is quasi-continuous. This is not a technical gap that a reader can fill; the definition of quasi-continuity fails exactly at boundary points of the open set. Since Theorem 2 uses this to derive L(X)≤t(QP(X)), Theorem 5 uses it to derive nw(X)≤ψ(QP(X,Y)), and Theorem 6 builds on both, the main positive results of Section 3 are unsupported. The same Lemma 2 is also misapplied in Theorem 3 with a union of open sets in the closed-set slot. These are internal inconsistencies in the arguments, not disagreements with prior consensus. The mapping section contains an additional failure: Theorem 8 asserts that every quasi-continuous function on an open subspace extends by a constant outside the subspace, which is false when the exterior has empty interior, e.g. X=R and Y the complement of a Cantor set; no open G contained in the exterior exists near a boundary point, so the extension fails quasi-continuity there. Since the proof text contains explicit false constructions and the paper does not provide an alternative route to the stated conclusions, rejection is the appropriate outcome. The counterexample for the characteristic function is elementary and would settle the concern on its own; it shows the claimed witnesses are not members of QP(R).","tokens_in":11118,"tokens_out":8540,"duration_ms":89064,"concrete_test":"Analytic check: take X=R and O=(0,1), and test f=1_O against the definition of quasi-continuity at x=0. With U=(-1,1) and V=(-1/2,1/2), every nonempty open G⊂U contains a point of O, where f=1, so f(G)⊄V; hence 1_O is not quasi-continuous. This directly disproves the construction used in Theorems 2 and 5. Also recompute the family {f_A} in Theorem 2 for an open cover of R; f_A takes value 1 on X - O_A, so the assertion that the zero function belongs to {f_A:A∈F'} is false. If the inequality is to be salvaged, a different quasi-continuous witness is required, and none is supplied in the paper.","verdict_should_be":"REJECT","load_bearing_attack":"Both Theorems 2 and 5 use the same unsupported construction. For an open set O (or V), the proof sets f=0 on O and f=1 on X - O and asserts f∈QP(X). This is false for arbitrary open O. If x∈∂O∩O, then for V=(1/2,2) containing f(x)=1, every nonempty open G⊂U meets X - O, where f=0, so f(G)⊄V; if x∈∂O∩(X - O), the symmetric choice V=(-1/2,1/2) meets points of O. Lemma 2 ([19, Lemma 4.12]) only prescribes a value at a single point and on a closed set; it cannot force values on an entire open set O_A. In Theorem 2, the constructed f_A satisfies f_A(O_A)=0 and f_A(X - O_A)=1, so f_A is not the zero function, yet the proof asserts 'zero function f0∈P' and uses the tightness approximation of f0 by elements of P; this does not follow. Theorem 5 repeats the characteristic-function step to build a network of X from a pseudocharacter family. Theorem 3 has a related misuse: it applies Lemma 2 with E=⋃J although ⋃J is open and need not be closed. Because these constructions are load-bearing for the main cardinal inequalities, the proof text does not establish the theorems as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":11427,"tokens_out":17902,"duration_ms":167920,"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":[{"comment":"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":"Section 3, Theorems 1, 2, and 5"},{"comment":"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":"Section 3, Theorem 2"},{"comment":"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":"Section 3, Theorems 3 and 4"},{"comment":"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":"Section 4, Theorem 8"},{"comment":"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":"Section 4, Theorem 11"},{"comment":"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.","section":"Section 3, Theorem 6"}],"minor_comments":[{"comment":"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":"Section 2, Preliminaries"},{"comment":"The symbol ω(Q_P(X,Y)) is used without definition; elsewhere the weight is denoted w(X). The notation should be unified.","section":"Section 2, displays (21) and (22)"},{"comment":"The word 'matrizable' should be 'metrizable'.","section":"Section 3, first paragraph"},{"comment":"The phrase 'It is oblivious π_Y(Q_P(X)) ⊂ Q_P(Y)' should read 'It is obvious that'.","section":"Section 4, Theorem 8 proof"},{"comment":"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":"Section 4, paragraph after Theorem 8"},{"comment":"Theorem 9 is stated without proof or explicit citation; a short proof or a reference should be supplied.","section":"Section 4, Theorem 9"},{"comment":"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.","section":"Section 5, Conclusion"}],"recommendation":"reject","confidential_remarks":"The manuscript addresses a legitimate topic, and Theorem 10's proof is sound, but the central theorems share a small set of repeated proof devices that violate the hypotheses of the cited lemmas. The errors are load-bearing and some are concrete (e.g., the extension construction in Theorem 8 fails for X=ℝ, Y=(0,1)). This is not a matter of polishing; the proofs would need to be substantially rewritten or the theorems weakened. I recommend rejection for the journal at this stage, though the topic itself could be suitable if the authors repair the arguments and carefully re-check the use of Lemma 2 and the quasi-continuity of characteristic functions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper is not ready for serious engagement. The main new claims—Theorems 2, 3, 5, 8, and 11—have load-bearing proof errors, and Theorems 8 and 11 are outright false. The paper does some things well: it organizes the Hola–Holy and Kumar–Tyagi results, and Theorem 10 (continuity of multiplication C_P(X)×Q_P(X)→Q_P(X)) has a correct epsilon argument. Those minor pieces are fine.\n\nThe core problem is systematic misuse of [19, Lemma 4.12], which only prescribes a value at a single point and on a closed set. Theorems 2 and 3 use it to force values on entire open sets (O_A and ⋃J), which the lemma cannot do. Theorem 5 uses the characteristic function of an arbitrary open set as if it were quasi-continuous; at boundary points it is not. These are not cosmetic gaps—they are the constructions that carry L(X)≤t(Q_P(X)), wc(X)≤ψ(Q_P(X,Y)), and nw(X)≤ψ(Q_P(X,Y)). Theorem 1's proof has a related issue: the step functions f_r jump on the boundary of U_r and are not quasi-continuous there.\n\nTheorem 8 (restriction map surjective/open onto Q_P(Y) for open Y) is false. For X=R, Y=R\\{0}, the quasi-continuous function on Y that is 0 left of 0 and 1 right of 0 has no quasi-continuous extension to R. The proof's extension by constant 1 on X\\Y fails at the boundary. Theorem 11 (dense image of r_*) is also false: take X=R, Y=R\\{0}, Z=R, and the continuous surjection r(x)=x for x>0, r(x)=x+1 for x<0. Let g(x)=2−3x. The basic open set around g with V_0=(1.5,2.5) and V_1=(−1.5,−0.5) contains g, but r_*(Q_P(R,Y)) misses it entirely—any f with r∘f in that set would have to be positive at 0 and negative at 1, and a quasi-continuous image of a connected space cannot change sign in R\\{0}. The proof's mistake is assuming the basic open set [x_i,U_i] in Q_P(X,Y) is nonempty; it may be empty.\n\nWho is this for? A specialist in C_p-theory could use the survey and the correct minor theorems, but the advertised extensions are not established. I would not cite it for the main results, and it does not deserve referee time as submitted. It needs substantial correction before it is worth a round of review.","headline":"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.","tokens_in":11960,"tokens_out":22325,"would_cite":false,"duration_ms":204305,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54C35","54A25","54C05","54C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The space of quasi-continuous functions mirrors the domain's cardinal size.","keywords":["function space","quasi-continuous functions","topology of point-wise convergence","weight","network weight","tightness","pseudocharacter","special maps"],"falsifier":"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 $Y$.","tokens_in":10935,"feed_emoji":"","tokens_out":11532,"duration_ms":90613,"temperature":0.7,"pith_summary":"This paper tries to show that the space of quasi-continuous functions with the topology of point-wise convergence inherits, or at least strongly constrains, several cardinal sizes of its domain. For an ordered Hausdorff space $X$, it claims the weight and network weight of $Q_P(X)$ both equal $|X|$; for regular $X$ and metric $Y$, the weak covering number of $X$ is a lower bound for the pseudocharacter of $Q_P(X,Y)$; and the network weight of $X$ is a lower bound for the pseudocharacter of $Q_P(X,Y)$. It also studies restriction and induced maps, claiming the restriction map is open, continuous, and surjective for open subspaces, and the induced map has dense image. If these results are correct, classical $C_p$-theory cardinal inequalities extend to the broader class of quasi-continuous functions, where the function space is no more complicated than the domain in these respects.","feed_headline":"Quasi-continuous function spaces match the domain's cardinal size","feed_subtitle":"For ordered Hausdorff domains, weight and network weight of the pointwise quasi-continuous function space both equal |X|.","key_machinery":"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.","core_discovery":"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 $X$.","pith_inferences":["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."],"forward_implications":["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 $Q_P(X,Z)$."],"supporting_citations":[{"why":"Supplies the prior result $w(Q_P(X))=|X|$ and the lemma (Lemma 4.2) used to build quasi-continuous step functions in Theorem 1.","marker":"[21]"},{"why":"Supplies Lemma 4.12, the separation lemma used to construct functions with prescribed values in Theorems 2, 3 and 4, along with denseness results used in Theorem 7.","marker":"[19]"},{"why":"Gives the equivalence of semi-continuity and quasi-continuity invoked in the discussion around the product and restriction results.","marker":"[5]"},{"why":"Provides the $C_p$-theory correspondence between tightness and Lindelöf degree that motivates Theorem 2.","marker":"[22]"},{"why":"Supplies standard $C_p$-theory facts about pseudocharacter, separability and network weight used in Theorems 4 and 6.","marker":"[9]"}],"fun_headline_variants":["Quasi-continuous function spaces match domain cardinalities","For ordered Hausdorff X, nw(Q_P)=w(Q_P)=|X|","Restriction maps open and surjective on Q_P spaces","Cardinal invariants of pointwise quasi-continuous spaces","Quasi-continuous spaces: weight and network weight equal |X|"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quasi-continuous function spaces match domain cardinalities","For ordered Hausdorff X, nw(Q_P)=w(Q_P)=|X|","Restriction maps open and surjective on Q_P spaces","Cardinal invariants of pointwise quasi-continuous spaces","Quasi-continuous spaces: weight and network weight equal |X|"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001142,"raw_usage":{"total_tokens":4723,"prompt_tokens":915,"completion_tokens":3808,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":3717}},"tokens_in":531,"tokens_out":3808,"duration_ms":26142,"temperature":1.0,"reasoning_tokens":3717,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:49:21.874160+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $Y$.","supporting_citations":[{"cited_title":"Topol.Appl","cited_arxiv_id":null,"evidence_quote":"Supplies the prior result $w(Q_P(X))=|X|$ and the lemma (Lemma 4.2) used to build quasi-continuous step functions in Theorem 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 4.12, the separation lemma used to construct functions with prescribed values in Theorems 2, 3 and 4, along with denseness results used in Theorem 7."},{"cited_title":"Real Anal","cited_arxiv_id":null,"evidence_quote":"Gives the equivalence of semi-continuity and quasi-continuity invoked in the discussion around the product and restriction results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $C_p$-theory correspondence between tightness and Lindelöf degree that motivates Theorem 2."},{"cited_title":"Springer, New York (2010)","cited_arxiv_id":null,"evidence_quote":"Supplies standard $C_p$-theory facts about pseudocharacter, separability and network weight used in Theorems 4 and 6."}],"review_version":1}