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Topological Nearly Entropy on Nearly Compact Spaces

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every nearly compact invariant subset carries the full entropy of the system on Hausdorff R-spaces.

desk verdict The paper's new definitions are fine but its central coincidence theorem has a load-bearing proof gap; the sup-of-entropy argument is invalid. read the letter →

arxiv 1908.02177 v1 pith:KD7KOCC5 submitted 2019-08-06 math.DS

Zabidin Salleh , Syazwani Gulamsarwar This is my paper · ORCID
classification math.DS MSC 54H2037B40
keywords topologicalnearlyentropycompactspaceR-mapR-dynamicalsystemR-spaceregularopencoverproduct
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 introduces a second version of topological entropy, Ent_n, defined for maps on nearly compact spaces, and proves it coincides with the previously defined Ent_N whenever the whole space is nearly compact. Its main result is that on a Hausdorff R-space — a space in which unions of regular open sets are again regular open — the entropy of a map equals the entropy of its restriction to any invariant nearly compact subset. If correct, the full dynamical complexity of such a system is already present on every invariant nearly compact piece, so adding points outside such a subset does not increase the entropy. The paper also establishes subadditivity inequalities for product systems and gives an explicit example with zero entropy.

What carries the argument

The core objects are the two mutually consistent entropy functions $\mathrm{Ent}_N$ and $\mathrm{Ent}_n$, built from the logarithm of the minimal cardinality of finite subcovers of iterated regular open joins. The load-bearing device is the cover-restriction equality of Theorem 4.3, which converts entropy of $f$ on an invariant subset $K$ into entropy of the restriction $f|_K$ on the subspace. The $R$-space hypothesis — that unions of regular open sets remain regular open — supplies the extension step that identifies the regular open covers of $K$ with restrictions of regular open covers of $X$, making the suprema interchangeable in the proof of Theorem 4.5.

What would settle it

The equality can be tested by finding a Hausdorff $R$-space $X$, an $R$-map $f$, and an invariant nearly compact subset $K$ with a regular open cover of $K$ that is not the restriction of any regular open cover of $X$; if such a cover has entropy exceeding that of every restrictable cover, then $\mathrm{Ent}_n(f|_K)$ can strictly exceed $\mathrm{Ent}_N(f)$, refuting the theorem.

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Extended reading notes

Core claim

The central claim is Theorem 4.5(b): if $(X,f)$ is a topological $R$-dynamical system, $X$ is Hausdorff and an $R$-space, and $K\in H(X,f)$, then $\mathrm{Ent}_N(f)=\mathrm{Ent}_n(f|_K)$. The argument rests on Theorem 4.3, which equates the two entropy notions with respect to a cover: $\mathrm{Ent}_N(f,\mathcal U,K)=\mathrm{Ent}_n(f|_K,\mathcal U|_K)$. The authors use the $R$-space property to extend every regular open cover of $K$ to a regular open cover of $X$ (adding $X\setminus K$), which allows them to interchange suprema and conclude that the supremum over invariant subsets equals the entropy of the restriction. The paper also shows $\mathrm{Ent}_n(f)=\mathrm{Ent}_N(f)$ for nearly compact $X$ and proves product inequalities for both entropy notions.

Load-bearing premise

The proof of the main equality assumes that every regular open cover of an invariant nearly compact subspace $K$ is the restriction of some regular open cover of $X$ with each set of the form $U_A\cap K$ and with $X\setminus K$ regular open; this extension property is stated without proof and is needed to equate the suprema over all covers of $X$ and of $K$.

Editorial extensions

If this is right

  • If Theorem 4.5(b) is correct, then on a Hausdorff $R$-space the topological nearly entropy of the whole system equals that of any invariant nearly compact subset, so entropy is fully localised on such pieces.
  • Theorem 4.4 unifies the two definitions: whenever $X$ itself is nearly compact, $\mathrm{Ent}_n(f)=\mathrm{Ent}_N(f)$, so the new definition is an extension rather than a competing notion.
  • The product inequality $\mathrm{Ent}_n(f\times h)\le \mathrm{Ent}_n(f)+\mathrm{Ent}_n(h)$ extends the classical subadditivity of topological entropy to the nearly compact setting, and the analogous $\mathrm{Ent}_N$ inequality holds on Hausdorff $R$-space products.
  • The example $f(x)=kx$ on $\mathbb R$ winds up with $\mathrm{Ent}_N(f)=0$, illustrating that the definition can detect simple dynamics as zero-entropy.
  • The product-space results suggest a route toward a generalised entropy for higher-dimensional nearly compact $R$-dynamical systems, preserving the classical subadditivity pattern.

Reading between the lines

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

  • If the equality holds for every invariant nearly compact subset, it suggests an analogue of the classical fact that entropy is supported on the non-wandering set; a natural test is whether nearly compact invariant sets are the only carriers of entropy in $R$-dynamical systems beyond Hausdorff $R$-spaces.
  • The $R$-space condition is strong enough that it may force every open set to be regular open; checking this could reveal that the theorem applies more broadly than its proof suggests, or that the condition can be replaced by a cover-extension axiom.
  • The zero-entropy example on $\mathbb R$ hints that nearly entropy may be insensitive to expanding maps on noncompact spaces; comparing $\mathrm{Ent}_N(f)$ and $\mathrm{Ent}_n(f)$ on a compactification would clarify whether the notion captures genuine complexity or only compactness effects.
  • The product inequalities are one-sided; testing equality for mixing systems on nearly compact spaces would show whether the subadditivity is strict, mirroring the classical situation for topological entropy.
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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

4 major / 5 minor

Summary. The paper introduces a new notion of topological nearly entropy, Entn, for R-dynamical systems on nearly compact spaces, and aims to relate it to the previously defined EntN for systems with invariant nearly compact subsets. The main results are: (i) Theorem 4.3, which claims an equality between Entn(f|K, U|K) and EntN(f, U, K) for K in H(X,f); (ii) Theorem 4.4, stating that when X itself is nearly compact, Entn(f)=EntN(f); and (iii) Theorem 4.5, which asserts that for Hausdorff R-spaces, EntN(f,K)=Entn(f|K,K) and, more strongly, EntN(f)=Entn(f|K) for every K in H(X,f). The paper also introduces the notion of R-space and proves several lemmas about it, and concludes with product-space inequalities for EntN and Entn.

Significance. If the main results were correct, they would extend Adler-Konheim-McAndrew topological entropy to a framework for nearly compact spaces via R-maps, and the equality in Theorem 4.5(b) would be a striking simplification: every invariant nearly compact subset would carry the full entropy of the system. The concept of R-space is potentially useful and the lemmas in Section 4 (Lemmas 4.1-4.3) are interesting in themselves. However, the central proofs contain serious gaps: Theorem 4.3 is not well-defined as stated, and Theorem 4.5(b) collapses a supremum over all invariant nearly compact subsets to a single fixed subset without justification. These issues affect the principal claims of the paper, so the contribution is not presently reliable.

major comments (4)
  1. [§4, Theorem 4.3] The quantity Entn(f|K, U|K) is not well-defined under the paper's own definitions. Definition 4.2 applies only when the underlying space is nearly compact and the cover is a regular open cover. In Theorem 4.3, K is only assumed to be nearly compact relative to X, which does not imply that K is nearly compact as a subspace; moreover, U|K = {U∩K : U∈U} need not be a regular open cover of K. For example, take X = R, K = [0,1] (which is nearly compact relative to R) and U = (0,2) (regular open in R). Then U∩K = (0,1], which is not regular open in the subspace K because int_K(cl_K((0,1])) = (0,1) ≠ (0,1]. Hence the proof of Theorem 4.3 applies the function Nn to a cover that may not be a regular open cover, and the claimed equality is unsupported.
  2. [§4, Theorem 4.5(b)] The proof of Theorem 4.5(b) contains a critical fallacy in the manipulation of suprema. After swapping the order of sup_K and sup_U, the proof writes 'By Theorem 4.3, EntN(f,U) = Entn(f|K,U|K)' and concludes that sup_U EntN(f,U) = Entn(f|K). But Theorem 4.3 is pointwise: it states that for a fixed K ∈ H(X,f), EntN(f,U,K) = Entn(f|K,U|K). It says nothing about the value of sup_{K∈H(X,f)} EntN(f,U,K) and does not show that the fixed K realizes this supremum or that Entn(f|K) is independent of K. The displayed chain of equalities would imply that every invariant nearly compact subset carries the full entropy, a much stronger statement that is not established and is generally false in the classical compact setting. This gap is load-bearing: without it, the equality EntN(f)=Entn(f|K) does not follow.
  3. [§4, Theorem 4.5(a), Eq. (4.1)] The derivation of equation (4.1) relies on unproved extension properties. The proof asserts that for every regular open cover U_K of the subspace K, there exists a regular open cover U of X such that A = U_A∩K for each A∈U_K, with X\K regular open, and then that the supremum over all such special covers U equals the supremum over all regular open covers of X. Neither assertion is proved, and Lemma 4.3 does not imply them. Moreover, the resulting restriction U|K contains the empty set (from X\K), which is not a regular open set, so U|K is not a regular open cover of K. The equality of the two suprema in (4.1) is therefore unjustified, and Theorem 4.5(a) is unsupported.
  4. [§4, Definition 4.2 and Theorem 4.5] The notation Entn(f|K, K) and Entn(f|K, UK, K) used in Theorem 4.5 is not defined anywhere. Definition 4.2 defines Entn(f, U) and Entn(f) only for a nearly compact space X. The proof then treats Entn as if it takes a space, a cover, and a subset as arguments. This is not merely a cosmetic issue: it obscures the fact that Entn for a restriction f|K requires the subspace K to be a nearly compact dynamical system in its own right, a property that is never established in the paper.
minor comments (5)
  1. [Title page] The title contains typographical errors: 'NEARL Y ENTROPY', 'COMP ACT', 'NEARL Y COMP ACT' should be corrected.
  2. [Abstract] The sentence 'the topological nearly entropy of f and it restriction f|K coincides' should read 'its restriction f|K'.
  3. [§4, Theorem 4.5(b) proof] The symbol 'EntN(f,U)' appears in the proof, but EntN is defined with three arguments: EntN(f,U,K). This should be 'EntN(f,U,K)'.
  4. [§5, Example 5.1] The claim that 'the only invariant compact subset of R and hence nearly compact subset of R is {0}' is misleading, because near compactness is weaker than compactness; the argument should justify why no other invariant nearly compact subset exists, or weaken the statement accordingly.
  5. [References] Some references contain typographical errors: [13] 'Intoduction' should be 'Introduction', and [9] 'Topoogical' should be 'Topological'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Ent_n is explicitly a special case of Ent_N; Theorem 4.5's gaps are unproved extension and sup-collapse claims, not circular reductions.

full rationale

The new notion Ent_n is not derived from Ent_N by construction; it is defined directly in Definition 4.2 and then shown in Theorem 4.4 to coincide with Ent_N when X is nearly compact. The key bridge, Theorem 4.3, is a genuine pointwise comparison: for a fixed invariant nearly compact subset K and a fixed regular open cover U, it proves Ent_N(f,U,K)=Ent_n(f|K,U|K) by constructing subcovers in both directions. This is not a tautology because the minimal-cardinality functions N_K and N_n are computed on different spaces, and the proof provides explicit inequalities both ways. The central claim of Theorem 4.5(b) is not circular, but it is incorrectly derived. After interchanging suprema, the proof writes 'By Theorem 4.3, EntN(f,U)=Entn(f|K,U|K)', whereas Theorem 4.3 is pointwise and does not imply that the single subset K realizes sup_K EntN(f,U,K). Separately, the proof asserts without proof that every regular open cover of K is the restriction U|K of a regular open cover U of X, an assertion needed for equation (4.1). These are mathematical gaps and correctness risks, not reductions of the conclusion to an input. The reliance on the authors' previous paper [9] supplies basic definitions and standard subadditivity facts; that self-citation is not load-bearing in the specific reduction sense. No fitted parameters, empirical predictions, or imported uniqueness theorems are involved, so no circular step is established.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main results rest on the new R-space property and on Hausdorff separation, plus two unstated regular-open extension assumptions that are never proved. There are no fitted numerical parameters. The paper's own prior results [9] are treated as background. The R-space definition is the most obviously paper-specific axiom, while the extension assumptions are hidden and potentially false in general.

assumptions (5)
  • ad hoc to paper X is an R-space (Definition 4.3): the union of any family of regular open subsets is regular open.
    Introduced solely to make the separation Lemma 4.1 and the coincidence Theorem 4.5 hold; not a standard topological property.
  • domain assumption X is Hausdorff in Theorems 4.5 and 5.2.
    Standard separation axiom assumed in the main results and used in Lemma 4.1.
  • domain assumption Every regular open subset of K is the trace on K of a regular open subset of X, and X\K is regular open (so that U' ∪ {X\K} is a regular open cover of X).
    Unproved and load-bearing in the proof of Theorem 4.5(a); no proof is given that regular open sets in the subspace K extend to X.
  • domain assumption For each regular open A_y in X×Y containing (x,y), there exist regular open neighborhoods U(x,y) of x and V(x,y) of y with U(x,y)×V(x,y)⊆A_y.
    Used in Lemma 5.5 to construct cover refinement; requires a regular open neighborhood base, a regularity property not stated in the paper.
  • standard math Theorems 2.1 and 2.2 of the authors' previous paper [9] (monotonicity and subadditivity of M_K and EntN).
    Cited prior results are taken as background; they are from the authors' own paper but are not proved here.

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Pith. "Pith review of Topological Nearly Entropy on Nearly Compact Spaces." pith.science (2026). https://pith.science/paper/KD7KOCC5

@misc{pith2026190802177,
  author       = {Pith},
  title        = {Pith review of: Topological Nearly Entropy on Nearly Compact Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KD7KOCC5}},
  note         = {Machine review of arXiv:1908.02177}
}
read the original abstract

In our previous paper [9], we have introduced topological nearly entropy, Ent_N (f) by restricting X into a class of nearly compact spaces. In the present paper, some additional properties of this notion are studied. Furthermore, we introduce another new notion of topological nearly entropy of f denoted by Ent_n (f) when the whole space X itself is nearly compact. We show the relationship between these two notions for the class of nearly compact subspaces. We also propose new space, namely, R-space in studying the topological nearly entropy on nearly compact and Hausdorff space. As a consequence, the topological nearly entropy of f and it restriction f|K coincides. Finally, some fundamental properties of topological nearly entropy for product space are obtained.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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