Softly $\pi g\hat{D}$-Normal Spaces

Fahed Zulfeqarr; M. C. Sharma; Neeraj Kumar Tomar

REVIEW 6 major objections 6 minor 28 references

Softly $\pi g\hat{D}$-Normal Spaces

T0 review · 6 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper introduces softly πgD̂-normal spaces and proves they form a topological property that is hereditary for closed domain subspaces, with a Urysohn lemma characterization.

desk verdict Standard template paper in generalized normality; novelty is nominal and the main hereditary/topological claims rest on unproved or false steps. read the letter →

arxiv 2506.08431 v1 pith:X3QU4JXN submitted 2025-06-10 math.GN

Neeraj Kumar Tomar , Fahed Zulfeqarr , M. C. Sharma This is my paper

classification math.GN MSC 54A0554C0854C1054D15

keywords πgD̂-openD̂-closedD̂-normalsoftlyalmostquasimildlycloseddomainsubspace

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

The paper defines a new weak separation property, softly πgD̂-normality, in which any disjoint pair consisting of one π-closed set and one regularly closed set can be separated by disjoint πgD̂-open sets. It then claims two structural results: the property is preserved by open continuous bijections, so it is a topological property, and it is inherited by closed domain subspaces. These results matter because they place the new class on the known ladder of near-normal spaces, between quasi-normality and mild normality, and they supply a Urysohn-type characterization. If the claims hold, any space obtained from a softly πgD̂-normal space by a homeomorphism or by passing to a closed domain subspace will keep the same separation capacity.

What carries the argument

The mechanism is the class of πgD̂-closed sets: a set A is πgD̂-closed if D̂cl(A) ⊆ U whenever A ⊆ U and U is π-open, where D̂cl is the closure operator built from D̂-closed sets. Complements are πgD̂-open. The paper uses this class to define softly πgD̂-normal spaces and to state Lemma 3.17, which says that intersecting a πgD̂-open set of X with a closed domain subspace M yields a πgD̂-open set of M. That restriction lemma is the load-bearing step in the proof that soft πgD̂-normality is hereditary for closed domain subspaces.

What would settle it

Find a closed domain subspace M of a softly πgD̂-normal space X and a πgD̂-open set A in X such that A∩M is not πgD̂-open in M; this would refute Lemma 3.17 and leave Theorem 3.18 unproved.

Watch

Extended reading notes

Core claim

The central claim is that softly πgD̂-normality is both a topological property and a hereditary property with respect to closed domain subspaces. More precisely, Theorem 3.14 asserts that a space is softly πgD̂-normal exactly when every disjoint pair consisting of a π-closed set and a regularly closed set can be separated by a continuous real-valued function into [0,1]. Theorem 3.15 and Corollary 3.16 show that an open continuous injective image of a softly πgD̂-normal space is softly πgD̂-normal, and Theorem 3.18 states that every closed domain subspace of a softly πgD̂-normal space is softly πgD̂-normal. The paper also introduces richer variants—πgD̂-normal, almost πgD̂-normal, quasi πgD̂-normal, and mildly πgD̂-normal—and gives preservation theorems relating these classes under continuous and irresolute maps.

Load-bearing premise

Lemma 3.17 is stated without proof: for a closed domain subspace M, intersecting a πgD̂-open set of X with M must remain πgD̂-open in M; if this restriction property fails, the hereditary theorem has no support.

Editorial extensions

If this is right

Closed and open (clopen) subspaces of a softly πgD̂-normal space are softly πgD̂-normal, since every clopen subset is a closed domain.
Every πgD̂-T3 space, defined as softly πgD̂-normal plus πgD̂-T1, is πgD̂-T2.
An open continuous injective image of a softly πgD̂-normal space is softly πgD̂-normal, so the property is invariant under homeomorphisms.
Several preservation theorems follow: continuous softly πgD̂-closed surjections, π-continuous πgD̂-closed surjections, and almost π-continuous almost πgD̂-closed surjections all preserve soft πgD̂-normality under suitable hypotheses.

Reading between the lines

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

If Lemma 3.17 is ever supplied with a proof, the heredity result would become fully rigorous, but the restriction to closed domain subspaces rather than arbitrary subspaces suggests the πgD̂-closure operator does not behave well under unrestricted restriction.
The Urysohn-type characterization points toward embedding theorems: a softly πgD̂-normal space may admit sufficiently many πgD̂-continuous real-valued functions to separate π-closed from regularly closed sets, a route the paper does not explore.
A natural test would be whether the heredity theorem characterizes closed domain subspaces: if some non-closed-domain subspace fails soft πgD̂-normality, then the restriction in the theorem is not merely technical.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Thanks for the report. I read the paper and your verdict is close to mine. The paper introduces a new class of generalized normal spaces by using πgD̂-closed sets. That is real but nominal: it is the same template as πgβ-normal or softly normal, with the generalized closed set replaced. The examples in Sections 2 and 3 do show the new classes are distinct. The advertised main results are not established. Theorem 3.12 assumes πgD̂-cl(U) is πgD̂-closed, but the intersection of πgD̂-closed sets need not be πgD̂-closed; no proof is given. Theorem 3.14 is an unproved Urysohn lemma, and the proof of Theorem 3.21 applies soft πgD̂-normality to singletons, which are not generally π-closed or regularly closed, so that separation argument does not go through. Theorem 4.5 contains a false equality: the argument gives f(F)⊂Y−V, not equality. Since Theorem 4.5 is used in the later preservation theorems, the error propagates. The heredity result, Theorem 3.18, rests entirely on Lemma 3.17, which is stated without proof. Passing to a closed domain subspace changes both the π-open sets and the D̂-closure operator, so the lemma is not automatic. It may be true, but the paper has to prove it. The topological property claim, Theorem 3.15, also depends on an unproved assertion about inverse images of π-closed subsets of f(X). So the two headline claims are unsupported as written. On the positive side, the definitions are coherent, the diagrams relating the various flavors are probably correct, and the authors cite the prior work honestly. The citation pattern is fine. Who would use this? Researchers cataloguing generalized normal spaces might find the definitions useful, but not with these gaps. The contribution is marginal and the errors are load-bearing. I would not send it to a referee. A desk reject with an invitation to resubmit after a careful re-proof is the right call. The authors need to fix Lemma 3.17, the closure property in Theorem 3.12, Theorem 4.5, and the singleton issue in 3.21 before this is credible.

Referee Report

6 major / 6 minor

Summary. The paper introduces a new class of generalized normal spaces, called softly πgD̂-normal spaces, defined by separating a π-closed set from a regularly closed set by disjoint πgD̂-open sets. It also defines related notions (almost, quasi, mildly, and πgD̂-normal spaces), gives examples and an implication diagram, and claims several characterizations, a Urysohn-type lemma, topological invariance, heredity for closed domain subspaces, and several preservation theorems. The central advertised results are that softly πgD̂-normality is a topological property (Corollary 3.16) and hereditary with respect to closed domain subspaces (Theorem 3.18).

Significance. If the advertised results were correct, the paper would add a new member to the family of generalized normality properties and would extend the soft-normality program of Sharma and Kumar. The paper is organized as a systematic ladder of related definitions and supplies numerous small examples, which is useful for orientation in this taxonomic area. However, the central claims are not established: several load-bearing proof steps are unproved, some are based on an apparent confusion between open sets and πgD̂-open sets, and at least one key subspace assertion is false. No machine-checked proofs or reproducible code accompany the paper. The positive contribution is currently only definitional and taxonomic, not a reliable set of theorems.

major comments (6)

[§3, Theorem 3.12] The proof of the equivalence (a)–(d) repeatedly uses the assertions that πgD̂cl(U) is πgD̂-closed and that πgD̂cl(X−W) = X−W for πgD̂-closed W (in steps a⇒b, b⇒c, and c⇒d). These assertions are never proved, and πgD̂cl is only defined as an intersection of πgD̂-closed sets without establishing that the class of πgD̂-closed sets is closed under arbitrary intersections or that πgD̂cl is a closure operator. Since every implication in Theorem 3.12 depends on these unproved identities, the characterization is unsupported.
[§3, Theorem 3.14] The Urysohn-type lemma is stated with no proof, only the remark that it can be proved in the usual way from Theorem 3.12. However, soft πgD̂-normality supplies disjoint πgD̂-open sets, not disjoint open sets, and no argument is given that converts πgD̂-open separations into a continuous real-valued function. The statement is load-bearing because it is presented as a new characterization, but the manuscript provides no bridge from πgD̂-open sets to ordinary open sets or to continuity.
[§3, Theorem 3.15 and Corollary 3.16] The proof of Theorem 3.15 asserts that soft πgD̂-normality of X gives 'two disjoint open sets U and V', but Definition 3.1 only gives disjoint πgD̂-open sets. The proof also uses the parenthetical before the theorem to claim that f^{-1}(A) is π-closed in X for A π-closed in f(X); that parenthetical concerns inverse images of subsets of Y, not of the subspace f(X). Thus the topological-invariance claim of Corollary 3.16 is not established.
[§3, Lemma 3.17 and Theorem 3.18] Lemma 3.17 is stated without proof and is the sole bridge used to prove heredity in Theorem 3.18. Moreover, Theorem 3.18 silently asserts that regularly closed and π-closed sets in a closed domain subspace M remain regularly closed and π-closed in X. This assertion is false in general: [0,1/2] is regularly closed in the closed domain subspace [0,1] of R, but it is not regularly closed in R. Therefore the advertised heredity result is not merely unproved; one of its key steps is invalid.
[§3, Theorem 3.21] The proof of Theorem 3.21 applies soft πgD̂-normality to the disjoint sets {x} and {y}, but Definition 3.1 requires one set to be π-closed and the other regularly closed. The only information available is that {x} is πgD̂-closed, which implies neither condition. Consequently the step is invalid and the theorem is unproved.
[§5, Theorems 5.1, 5.5, 5.6, and 5.7] Several preservation theorems apply soft πgD̂-normality to pairs of sets that do not satisfy the definition's asymmetry condition. For example, Theorems 5.5–5.7 start with two disjoint π-closed sets (or two closed sets in Theorem 5.1), while soft πgD̂-normality only separates a π-closed set from a regularly closed set. Theorem 5.1 also uses the undefined term 'softly πgD̂-closed function' and contains the notation f^{-1}(V1) before V1 is defined. Theorem 5.7 uses almost π-continuity to infer that preimages of π-closed sets are closed, although Definition 4.1(7) only covers inverse images of regular closed sets. These preservation claims are therefore unsupported.

minor comments (6)

[§3, Definition 3.20] The term πgD̂-T1 is used in Theorem 3.21 but is never defined before that point.
[§5, Theorem 5.1] The proof contains a typographical error: f^{-1}(V1) appears before V1 is introduced; it should presumably be f^{-1}(M1) and f^{-1}(M2).
[§5, Theorem 5.6] The proof says 'Let G = int(cl(V)) and H = int(cl(V))'; the second should be int(cl(U)). Also 'π' appears where 'φ' is intended.
[§4, Theorem 4.5] The proof states that 'f(F) is ĝD-closed' but the context requires πgD̂-closed. The equality f(F) = Y−V is valid only after explicitly using surjectivity, which should be stated.
[§2, Example 2.15] The example lists the g-closed sets of X={a,b,c} as including {a,c,d} and {a,b,c,d}, which are not subsets of X; this appears to be a copying error and should be corrected.
[§4 and §5] The manuscript uses several undefined or unexplained phrases, such as 'softly πgD̂-closed' (Theorem 5.1) and 'disjoint softly πgD̂-open sets' (Theorem 5.6), which should be aligned with the definitions given in Section 4.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the new class is defined from πgD̂-open sets, not from the conclusion, and the main theorems are direct proofs; the unproved Lemma 3.17 is a correctness gap, not a circular step.

full rationale

The derivation chain is self-contained with respect to its own definitions. The paper defines πgD̂-closed sets from the previously defined D̂cl operator (Definitions 2.3-2.5) and then defines softly πgD̂-normal as a separation property (Definition 3.1). The equivalences in Theorems 3.12 and 3.13 are proved by elementary set manipulations from that definition; no equation is identified with its own input, and no result is assumed as the very statement being derived. The topological-property claim (Theorem 3.15 and Corollary 3.16) and the preservation theorems are direct diagram chases from the definitions. The only author self-citation is Reference [11] (M. C. Sharma is an author) for the base notion 'softly normal'; the new class is not a restatement of [11], and the main theorems do not depend on a result imported from that citation. Two passages are genuine gaps but they are not circularity: Lemma 3.17 is asserted without proof and Theorem 3.18 depends entirely on it, and the restriction of πgD̂-openness to a closed domain subspace is not automatic because D̂cl and the family of π-open supersets both change under subspaces. Similarly, Theorem 3.14's Urysohn direction is delegated to 'a similar way of the normal case' without proof. These are omitted proofs or possible correctness failures, not reductions of a claim to its own input. There are no fitted parameters, no empirical predictions, no imported uniqueness theorems, and no ansatz smuggled in through a citation. The modest score reflects only the one non-load-bearing self-citation for the base definition.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or empirical quantities appear; the paper is a purely definitional taxonomy. The new definitions (πgD̂-closed sets, softly πgD̂-normal spaces) are the paper's stated contribution rather than unsupported empirical postulates, so they are not listed as invented entities. The load-bearing assumptions are the axioms above: the πgD̂-closure operator is treated as a closure operator without proof, restriction to closed domain subspaces is assumed without proof, and soft normality is applied outside its definitional scope.

assumptions (4)

ad hoc to paper πgD̂cl(A) is πgD̂-closed for every A, and πgD̂cl(X-W)=X-W for πgD̂-closed W.
Invoked in Theorem 3.12(b=>c) and (c=>d); no proof is supplied that the intersection of πgD̂-closed sets is πgD̂-closed.
ad hoc to paper If M is a closed domain subspace of X and A is πgD̂-open in X, then A∩M is πgD̂-open in M.
Lemma 3.17, unproved, used in Theorem 3.18.
domain assumption Inverse images of regular closed and π-closed sets under an open continuous function are respectively regular closed and π-closed.
Asserted before Theorem 3.15 and used to transfer soft πgD̂-normality to f(X).
ad hoc to paper Soft πgD̂-normality can separate two arbitrary singletons.
Theorem 3.21 applies Definition 3.1 to {x} and {y}, which are not known to be π-closed or regular closed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Softly $\pi g\hat{D}$-Normal Spaces." pith.science (2026). https://pith.science/paper/X3QU4JXN

@misc{pith2026250608431,
  author       = {Pith},
  title        = {Pith review of: Softly $\pi g\hatD$-Normal Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X3QU4JXN}},
  note         = {Machine review of arXiv:2506.08431}
}

abstract

The aim of this paper is to introduce a new class of softly normal called softly $\pi g\widehat{D}$ -normality by using $\pi g\widehat{D}$ -open sets and obtained several properties of such a space. We discuss many properties of this new space and we give some properties that connect this new spaces with some other topological spaces, also we present some examples and counter examples that show the relationships between softly $\pi g\widehat{D}$ -normal spaces and some other topological spaces, also we introduced the concept of $\pi g\widehat{D}$ -normal, almost $\pi g\widehat{D}$ -normal, quasi $\pi g\widehat{D}$ -normal, mildly $\pi g\widehat{D}$ -normal. The main result of this paper is that softly $\pi g\widehat{D}$ -normality is a topological property and it is a hereditary property with respect to closed domain subspaces. Moreover, we obtain some new characterizations and preservation theorems of softly $\pi g\widehat{D}$ -normal spaces. We insure that existence of utility for new results of softly $\pi g\widehat{D}$ -normality using separation axioms in topological spaces which is separate on a known separation axioms in topological spaces.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Zaitsev, V., On certain classes of topological spaces and their biocompa ctiﬁcations, Dokl. Akad. Nauk SSSR, 178, 778-779 (1968)

work page 1968
[2]

Singal, M. K. and Arya, S. P., Almost normal and almost completely regular spaces , Glasnik Mathematicki Tom, 5(25), 1, 141-152 (1970)

work page 1970
[3]

V., Real functions and near normal spaces , Sibirskii Mat

Shchepin, E. V., Real functions and near normal spaces , Sibirskii Mat. Zhurnal, 13, 1182-1196 (1972)

work page 1972
[4]

Singal, M. K. and Singal, A. R., Mildly normal spaces , Kyungpook Math. J., 13, 27-31 (1973)

work page 1973
[5]

and Rahman, M

Lal, S. and Rahman, M. S., A note on quasi-normal spaces , Indian J. Math., 32, 87-94 (1990)

work page 1990
[6]

N., π-normal toplogical spaces , Filomat, 22 (1), 173-181 (2008)

Kalantan, L. N., π-normal toplogical spaces , Filomat, 22 (1), 173-181 (2008)

work page 2008
[7]

Thabit, S. A. S. and Kamaruihaili, H., π-normality, weak regularity and the product of topo- logical spaces, European Journal of Scientiﬁc Research, 51(1), 29-39 (2011)

work page 2011
[8]

Thabit, S. A. S. and Kamaruihaili, H., πp-normality on topological spaces, Int. J. Math. Anal., 6(21),1023-1033 (2012)

work page 2012

Show all 28 references

[9]

G., Benchalli, S

Patil, P. G., Benchalli, S. S. and Gonnagar, P. K., ωα -separation axioms in topological spaces, Jour. of New Results in Science, 5, 96-103 (2014)

2014
[10]

and Rajpal, R., πgβ -normal spaces in topological spaces, International J

Hamant, K., Umesh, C. and Rajpal, R., πgβ -normal spaces in topological spaces, International J. of Science and Research, 4(2), 1531-1534 (2015)

2015
[11]

M. C. Sharma and Hamant Kumar, Softly normal topological spaces , Acta Ciencia Indica, Vol. XLIM no. 2, 81-84 (2015)

2015
[12]

M. H. Stone, Applications of the theory of Boolean rings to general topol ogy, Trans. Amer. Math. Soc., 41(1937), 375-381

1937
[13]

A. S. Mashhour, M. E. Abd. El- Monsef and S. N. EL- Deeb, On pre continuous mappings and weak pre- continuous mappings , Proc. Math, Phys. Soc. Egypt., 53(1982), 47-53

1982
[14]

Levine, Semi-open sets and semi-continuity in topological spaces , Amer

N. Levine, Semi-open sets and semi-continuity in topological spaces , Amer. Math. Monthly, 70 (1963), 36-41

1963
[15]

On Some Class of Nearly Open Sets,

O. Njastad, “On Some Class of Nearly Open Sets,” Paciﬁc. J. Math., vol. 15, no. 3, pp. 961-970, 1965

1965
[16]

Andrijevie, Semi- preopen sets , Math

D. Andrijevie, Semi- preopen sets , Math. Vesnik, 38(1986), 24-32

1986
[17]

Levine, generalized closed sets in topology , Rend

N. Levine, generalized closed sets in topology , Rend. Circ. Mat. Palermo, 19 (2) (1970), 89-96

1970
[18]

Quasi Normal Spaces and πg -closed sets

J.Dontchev and T.Noiri, “Quasi Normal Spaces and πg -closed sets”, Acta Math. Hungar.,89, 3, (2000), 211-219. 12

2000
[19]

Maki, R, Deviand K

H. Maki, R, Deviand K. Balachandran, Generalized α - closed sets in topology , Bull. Fukuka Univ. Ed. Part. III 42(1993) 13-21

1993
[20]

Studies on πgα -closed sets in Topology

C.Janaki, “Studies on πgα -closed sets in Topology” , Ph.D Thesis, Bharathiar University, Coimbatore.(2009)

2009
[21]

Dontchev, On generalized semi-pre open sets , Mem

J. Dontchev, On generalized semi-pre open sets , Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 16(1995), 35-48

1995
[22]

Palaniappan and K

N. Palaniappan and K. C. Rao, Regular generalized closed sets , Kyungpook, Math. J., 33(1993), 211-219

1993
[23]

Gnanambal, On generalized pre-regular closed sets in topological spac es, Indian J

Y. Gnanambal, On generalized pre-regular closed sets in topological spac es, Indian J. Pure Appl. Math., 28(1997), 351-360

1997
[24]

Sundaram P and Sheik john M., On α -closed sets in topology , Acta Ciencia Indica, 4(2000), 389-392

2000
[25]

M. K. R. S. Veera Kumar, Between closed sets and g closed sets , Mem. Fac. Sci. Kochi Univ. (Math.), 21(2000), 1-19

2000
[26]

J.Antony Rex Rodrigo and K.Dass, A New type of generalized closed sets , IJMA, 3(4)(2012), 1517-1523

2012
[27]

Dass and G

K. Dass and G. Suresh, ˆD- closed set in topological spaces , MJM, vol. s. No. 1, (2021) 265- 269

2021
[28]

Noiri, Mildly - normal spaces and some functions

T. Noiri, Mildly - normal spaces and some functions . Kyungpook Math. J.,36(1996), 183 -190. 13

1996

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Zaitsev, V., On certain classes of topological spaces and their biocompa ctiﬁcations, Dokl. Akad. Nauk SSSR, 178, 778-779 (1968)

work page 1968

[2] [2]

Singal, M. K. and Arya, S. P., Almost normal and almost completely regular spaces , Glasnik Mathematicki Tom, 5(25), 1, 141-152 (1970)

work page 1970

[3] [3]

V., Real functions and near normal spaces , Sibirskii Mat

Shchepin, E. V., Real functions and near normal spaces , Sibirskii Mat. Zhurnal, 13, 1182-1196 (1972)

work page 1972

[4] [4]

Singal, M. K. and Singal, A. R., Mildly normal spaces , Kyungpook Math. J., 13, 27-31 (1973)

work page 1973

[5] [5]

and Rahman, M

Lal, S. and Rahman, M. S., A note on quasi-normal spaces , Indian J. Math., 32, 87-94 (1990)

work page 1990

[6] [6]

N., π-normal toplogical spaces , Filomat, 22 (1), 173-181 (2008)

Kalantan, L. N., π-normal toplogical spaces , Filomat, 22 (1), 173-181 (2008)

work page 2008

[7] [7]

Thabit, S. A. S. and Kamaruihaili, H., π-normality, weak regularity and the product of topo- logical spaces, European Journal of Scientiﬁc Research, 51(1), 29-39 (2011)

work page 2011

[8] [8]

Thabit, S. A. S. and Kamaruihaili, H., πp-normality on topological spaces, Int. J. Math. Anal., 6(21),1023-1033 (2012)

work page 2012