{"id":"fafd3bf4-5533-4242-9f4c-b39525275c6d","arxiv_id":"1908.02845","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Hyperspaces of countable compacta with bounded numbers of accumulation points are homeomorphic to Q^ω, S4, c0, or H(I), depending on the ambient space.","lead":"This paper characterizes several hyperspaces of countable compact subsets of metric spaces: the space of all countable compacta, and subspaces of compacta with at most n, finitely many, or countably many accumulation points. It proves that for many natural spaces these hyperspaces are homeomorphic to standard infinite-dimensional spaces such as products of rationals, the space c0, and the Hurewicz set of the interval.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The homeomorphism classifications in Theorems 8.5–10.1 hinge on Theorem 5.3, quoted from the first author's unpublished preprint [2]; if that theorem has hidden hypotheses or a proof gap, the absorbing-set uniqueness step has no support.","rationale":"Agree with the reader that Theorem 5.3 is the weakest assumption. The alternative candidate, Lemma 6.3, is a genuine proof error but local: the claimed inclusion is wrong, yet Sigma4 is still contained in the pseudo-boundary of the Hilbert cube, a sigmaZ-set, so Corollary 6.4 can be recovered without changing the main argument. The central homeomorphisms A_omega(X) homeomorphic to Sigma4 and H(X) homeomorphic to H(I) require A_omega(X) and H(X) to be absorbers, which in turn requires strong universality of the relevant pairs. That strong universality is obtained only through Theorem 5.3, a result quoted from an unpublished preprint of one of the authors. A theorem with the entire classification depending on it, and with the only stated proof residing in an inaccessible preprint, is a legitimate condition for acceptance. The remainder of the paper is detailed and Sections 9-10 contain independent constructive arguments for A_n(I) and A_n(S1), so a conditional accept rather than a reject is justified. The concern is purely about verification of the cited theorem, not about the authors' integrity. Thus the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":23761,"tokens_out":24515,"duration_ms":265861,"concrete_test":"Obtain [2] and check that Theorem 9's hypotheses are exactly those stated in Theorem 5.3 (dense coideal, LC0, Pi^0_2-hereditary class) with no additional Polishness, SDAP, or subsemilattice conditions. Then prove Theorem 8.3 in the special case M = X = I, C = Sigma^0_4 by explicitly constructing the strong-universality Z-embeddings for the pair (K(I), A_omega(I)), using the map from Lemma 8.1 and the homotopy-density deformation from Theorem 7.3. If the direct construction succeeds, the dependency is benign; if a missing hypothesis or a gap emerges, Theorem 8.5's classification of A_omega(X) and H(X) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Reading in good faith, the core theorems are well prepared: the descriptive-complexity results, the zero-dimensional cases, and the explicit constructions for A_n(I) in Section 9 stand on their own. The load-bearing point is the step from Lemma 8.2 (everywhere preuniversality) to Theorem 8.3 (strong universality). That step is Theorem 5.3, asserted to be a special case of Theorem 9 of the first author's preprint [2] but not proved here. Theorem 8.4 turns Theorem 8.3 plus a sigmaZ-cover into absorbing-pair and absorbing-space conclusions, and Theorem 8.5 uses Theorem 4.13 or 4.5 to identify A_omega(X) with Sigma4 and H(X) with H(I). If Theorem 5.3 needs an extra hypothesis — for example that the semilattice M is Polish and X has SDAP, or that X is a subsemilattice rather than merely a coideal, or that C is Wadge-closed rather than only Pi^0_2-hereditary — then the implication from everywhere preuniversality to strong universality may fail and the uniqueness conclusion collapses. No internal inconsistency is visible, but this is the least protected part of the argument. A separate local flaw is Lemma 6.3: the displayed inclusion Sigma4 subset union_i X_i with X_i = {x_n = 0 for n <= i} is false. However, every point of Sigma4 has infinitely many zero coordinates, hence lies in the pseudo-boundary of I^omega, which is a sigmaZ-set; so the lemma's conclusion is repairable and is not the main threat.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies hyperspaces of countable compact subsets of metric spaces: the spaces H(X) of all countable compacta, A_n(X) of infinite compacta with at most n accumulation points, A_omega(X) with finitely many, and A_{omega+1}(X) with countably many accumulation points. The main contributions are: (i) Borel complexity estimates for these hyperspaces, including sharpness results via the standard sets P_3 and S_4; (ii) topological characterizations in the zero-dimensional case, e.g., A_n(X) is homeomorphic to Q^omega and A_omega(X) to S_4 for dense-in-themselves 0-dimensional Polish or sigma-compact spaces; and (iii) absorbing-set classifications for connected locally connected spaces: A_omega(X) is homeomorphic to Sigma_4, while A_{omega+1}(X) and H(X) are homeomorphic to H(I); additionally A_n(I) and A_n(S^1) are homeomorphic to c_0. The proofs combine descriptive set theory with the theory of strongly universal and absorbing sets in infinite-dimensional topology.","tokens_in":24093,"tokens_out":11642,"duration_ms":129283,"significance":"If the results are correct, the paper gives a nearly complete topological classification of the principal hyperspaces of countable compacta for large natural classes of spaces. The main theorems extend Cauty's characterization of H(I), answer questions from the literature about convergent-sequence hyperspaces, and introduce explicit Sigma_4 and Pi_3 absorbers in the Hilbert cube that are likely to be of independent use. The zero-dimensional characterizations in Sections 3 and 7 rest on classical theorems and appear solid. The paper is carefully structured, with detailed constructions for the interval and circle cases. The debt to the first author's unpublished preprint [2] for the key bridge from preuniversality to strong universality is the principal weakness; the false inclusion in Lemma 6.3 is a local error whose conclusion is nevertheless repairable. Overall, the architecture of the proof is compelling, but the load-bearing external theorem needs to be made fully verifiable.","major_comments":[{"comment":"Theorem 5.3 is the central bridge from everywhere preuniversality to strong universality for dense coideals in Lawson semilattices, and it is cited to the first author's unpublished preprint [2, Theorem 9]. This theorem is used in Theorem 8.3 to lift Lemma 8.2 to strong universality, and hence underpins Theorems 8.4, 8.5, and the absorbing-set classifications in Sections 9-11. Since [2] is not publicly available, the reader cannot check hypotheses such as the precise meaning of 'Pi^0_2-hereditary' for the classes Pi^0_3, Sigma^0_4, and Pi^1_1, nor whether the coideals considered (A_omega(X), A_{omega+1}(X), H(X), and An(X)) satisfy all conditions. This is a load-bearing point. I recommend that the authors either prove Theorem 5.3 in the paper or replace the citation by a precise statement of a published version; alternatively, they should verify directly, in the present paper, that the specific pairs used satisfy the needed strong universality.","section":"Section 5, Theorem 5.3"},{"comment":"The displayed inclusion in Lemma 6.3, Sigma_4 subset union_{i in omega} X_i with X_i = {(x_n) in I^omega : x_n = 0 for n <= i}, is false. For example, take x in I^omega with x_{2i+1}=1 for all i and x_{2i(2k+1)}=0 for all i,k large enough; then x belongs to Sigma_4 because the defining condition imposes zeros only on the sparse set {2i(2k+1)}, but x is not in any X_i since every X_i forces all coordinates up to i to be zero, while x has arbitrarily large odd coordinates equal to 1. The lemma's conclusion is nevertheless correct: every point of Sigma_4 has infinitely many zero coordinates, hence lies in the pseudo-boundary of I^omega, which is a sigmaZ-set. The proof should be corrected by using this pseudo-boundary argument rather than the false inclusion. As the lemma is used in Corollary 6.4 for the absorbing-pair conclusions, the correction must be made, but it does not undermine the main results.","section":"Section 6, Lemma 6.3"},{"comment":"The step from the pair (K(M), A_omega(X)) being strongly universal to the space A_omega(X) being strongly universal is not automatic and depends on the second part of Theorem 5.3, which requires X to have SDAP. In Theorem 8.4 the authors do invoke Fact 4.3 and Fact 4.12, but in Theorem 9.3 the passage from the absorbing pair (K(I), A_n(I)) to the absorbing space A_n(I) is stated without explicitly verifying the hypotheses of Fact 4.12, namely that A_n(I) has SDAP and is homotopy dense in K(I). The latter does follow from Fact 4.7 once the pair is strongly universal and K(I) is an ANR, but this chain of implications should be spelled out for the reader, especially because A_n(X) is not a subsemilattice and is not covered by the earlier semilattice arguments.","section":"Section 8, Theorem 8.3 and Section 9, Theorem 9.3"}],"minor_comments":[{"comment":"The hypothesis says 'for any nonempty set U subset M' but the proof and usage require U to be open; replace 'set' by 'open set'.","section":"Section 8, Lemma 8.2"},{"comment":"The statement 'U is an nonempty open subset of E' appears to contain a typo: E is not defined; it should presumably be 'of M' or 'of X'.","section":"Section 4, Fact 4.9"},{"comment":"The citation [14] for Cauty's result that (K(I), H(I)) is Pi^1_1-absorbing seems to point to Curtis and Schori, not to Cauty's paper; reference [7] is the intended Cauty citation.","section":"Section 8, Lemma 8.1"},{"comment":"There are several minor typos and OCR artifacts, e.g., 'den se-in-itself' in the abstract and 'characterzations' in the heading of Section 8; these should be corrected in a final pass.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The conditional verdict of the reader is appropriate. The main theorems are plausible and the proof architecture is sound, but the dependence on the unpublished preprint [2] for Theorem 5.3 is a serious verifiability issue. The false inclusion in Lemma 6.3 is easily repaired and should not by itself drive the decision. I would recommend that the editors require the authors to either include a full proof of Theorem 5.3 in an appendix, or cite a published version, before acceptance. Given the scope of the paper, this is a major revision rather than a rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take on Banakh–Krupski–Omiljanowski, “Hyperspaces of countable compacta.” The paper is a substantial piece of work. It settles the open questions from [20] and [6], and gives complete topological classifications for the hyperspaces A_n(X), A_omega(X), A_{omega+1}(X), and H(X) for large classes of Polish spaces. The descriptive-complexity part (Section 2) and the zero-dimensional characterizations (Section 3) stand on their own. The later sections show, among other things, that A_n(I) and A_n(S1) are homeomorphic to c0, that A_omega(X) is homeomorphic to Sigma4, and that H(X) is homeomorphic to H(I) for connected locally connected Polish spaces that are locally compact or nowhere locally compact. The constructions of Sigma4 and Pi3 as absorbers are new, and the paper says so honestly.\n\nThe proofs are detailed and mostly check out. I have two caveats.\n\nFirst, Lemma 6.3 contains a false inclusion: it claims Sigma4 is contained in the union of X_i = {x_n = 0 for n <= i}. That is not true. But the lemma's conclusion is still right, because every point of Sigma4 has infinitely many zero coordinates, so Sigma4 sits inside the pseudo-boundary of I^omega, which is a sigma-compact sigmaZ-set. So this is a local mistake, not a structural one. It should be fixed, but it does not threaten the main theorems.\n\nSecond, and more importantly, the step from everywhere preuniversality to strong universality is Theorem 5.3, quoted from the first author's unpublished preprint [2]. That theorem is load-bearing for the absorbing-set arguments in Sections 8–11. If it has hidden hypotheses or a proof gap, the uniqueness results for A_omega(X) and H(X) lose their foundation. I have no reason to think it is false, but I cannot verify it from the paper. A referee should ask for the preprint or an appendix proof.\n\nThere are also minor typos in Section 3 (e.g., ‘Aw(X)’ for A_omega(X)), but nothing confusing.\n\nAll in all, this is a serious paper by people who know the field. The central claims are plausible and well supported. It deserves peer review. I would send it to a good topology journal and ask the authors to fix Lemma 6.3 and clarify the status of Theorem 5.3.","headline":"A substantial, mostly sound paper that settles open questions in hyperspace theory; the main caveat is a load-bearing theorem quoted from an unpublished preprint, plus a repairable false inclusion in Lemma 6.3.","tokens_in":24668,"tokens_out":3311,"would_cite":true,"duration_ms":32873,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57N20","54B20","54H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper classifies, up to homeomorphism, the hyperspaces of countable compact subsets of a metric space and their accumulation-point strata, showing that zero-dimensional dense-in-itself Polish spaces produce only the models $Q^\\omega$…","keywords":["hyperspace","countable compacta","accumulation points","absorbing sets","Hilbert cube","Borel complexity","c0","Peano continuum"],"falsifier":"Exhibit a Lawson semilattice $M$ and a dense coideal $X$ that is locally path-connected in $M$ for which the pair $(M,X)$ is everywhere preuniversal but not strongly universal for some $\\Pi^0_2$-hereditary class $C$; that directly refutes the equivalence theorem used in Theorem 8.3 and would invalidate the absorbing-set route to Theorem 8.5. A less surgical test: find two nondegenerate connected locally connected Polish spaces, one locally compact and one nowhere locally compact, whose hyperspaces $H(X)$ and $H(Y)$ are not homeomorphic.","tokens_in":23554,"feed_emoji":"🧊","tokens_out":11743,"duration_ms":113074,"temperature":0.7,"pith_summary":"This paper classifies, up to homeomorphism, the hyperspaces of countable compact subsets of a metric space: the full space $H(X)$, the space $A_n(X)$ of infinite compacta with at most $n$ accumulation points, and the variants with finitely or countably many accumulation points. For dense-in-itself 0-dimensional Polish spaces it proves the $n$-th level is always homeomorphic to $Q^\\omega$, the finite-accumulation level to the standard $\\Sigma^0_4$-complete set $S_4$, and the top levels to $H(Q)$. For nondegenerate connected locally connected Polish spaces that are either locally compact or nowhere locally compact, it proves $A_\\omega(X) \\cong \\Sigma_4$ and $A_{\\omega+1}(X) \\cong H(X) \\cong H(I)$, where $\\Sigma_4$ is a concrete $F_{\\sigma\\delta\\sigma}$-absorber in the Hilbert cube constructed in the paper. For intervals and the circle it shows $A_n(X) \\cong \\Pi_3 \\cong c_0$, the space of real sequences converging to 0. The interest is that these natural and initially complicated spaces turn out to have a handful of fixed topological types determined by very coarse features of $X$.","feed_headline":"Few fixed shapes classify countable-compacta hyperspaces","feed_subtitle":"Rationals, intervals, and circles each produce hyperspaces of fixed, known topological type.","key_machinery":"The engine is the theory of absorbing sets in infinite-dimensional manifolds. An absorber for a Borel class $C$ is a space in $C$ that is strongly $C$-universal and is a countable union of Z-sets belonging to $C$; such spaces are unique up to homeomorphism. The paper builds two concrete absorbers inside the Hilbert cube $I^\\omega$: $\\Pi_3$, an $F_{\\sigma\\delta}$ set that is later identified with $c_0$, and $\\Sigma_4$, an $F_{\\sigma\\delta\\sigma}$ set, and proves the pairs $(I^\\omega, \\Pi_3)$ and $(I^\\omega, \\Sigma_4)$ are absorbing for the classes $\\Pi^0_3$ and $\\Sigma^0_4$. The bridge from hyperspaces to these absorbers uses the Vietoris hyperspace $K(X)$ as a Lawson semilattice under union; a cited theorem converts the easier-to-check everywhere preuniversality of a pair $(K(M), A(X))$ into strong universality, after which absorbing-pair uniqueness produces the homeomorphisms. The 0-dimensional results run through separate machinery: classical homeomorphism criteria for zero-dimensional first-category spaces in a given absolute Borel class.","core_discovery":"On the paper's own terms, the central discovery is that the topological type of a hyperspace of countable compacta is determined by the descriptive class of the underlying space's derived-set structure, not by its local geometry. Theorem 3.5 identifies $A_n(X)$ with $Q^\\omega$ for dense-in-itself 0-dimensional Polish $X$, $A_\\omega(X)$ with $S_4$, and $A_{\\omega+1}(X)$ and $H(X)$ with $H(Q)$; Theorem 3.7 gives the same models for pointed hyperspaces $A_n(X,F)$ and $A_\\omega(X,F)$ over 0-dimensional $\\sigma$-compact metric spaces. The infinite-dimensional half of the paper, Theorem 8.5, asserts that for nondegenerate connected locally connected Polish spaces $X$ that are locally compact or nowhere locally compact, $A_\\omega(X)$ is homeomorphic to the absorber $\\Sigma_4$ and $A_{\\omega+1}(X)$ and $H(X)$ are homeomorphic to $H(I)$. Theorem 9.3 and Theorem 10.1 then give $A_n(I) \\cong A_n((0,1)) \\cong A_n(S^1) \\cong \\Pi_3 \\cong c_0$ for every $n \\in \\mathbb{N}$, and Theorem 11.1 extends the $c_0$ model to $A_1(X,\\{p\\})$ at a point $p$ of order $\\ge 2$ in a Peano continuum. Read sympathetically, these theorems fully settle the topological classification of the principal hyperspaces of countable compacta for the stated classes of underlying spaces.","pith_inferences":["A natural test the paper does not run: take a connected locally connected Polish space that is neither locally compact nor nowhere locally compact, such as a space with one compact component and one noncompact component, and check whether $A_\\omega(X)$ is still $\\Sigma_4$ or whether the two-regime hypotheses in Theorem 8.5 are essential.","The paper reduces the longstanding $A_1(\\mathbb{R}\\setminus\\mathbb{Q})$ versus $A_1(\\mathbb{Q})$ question to whether $A_1(\\mathbb{Q})$ is an absolute $F_{\\sigma\\delta}$ set; computing that absoluteness is a concrete next experiment that would either extend the 0-dimensional classification or expose its boundary.","The Section 11 technique is tied to arcs by a continuous selection of a point from a finite set, and the paper notes this selection is characteristic of arcs; a companion conjecture suggested by the results is that for Peano continua with no point of order 2, the $c_0$ model survives only under the order $\\ge 2$ hypothesis.","If the underlying equivalence theorem for Lawson semilattices were weakened, the main homeomorphism claims would likely still hold for the zero-dimensional cases, since those rely on different classical criteria; this separation could be used to isolate which results actually depend on the cited preprint theorem."],"forward_implications":["For intervals and the circle, $A_n(X)$ is homeomorphic to $c_0$ for every $n$, so the topological type carries no information about $n$ or about whether endpoints are included.","For any nondegenerate connected locally connected Polish space $X$ that is locally compact or nowhere locally compact, $A_{\\omega+1}(X)$ and $H(X)$ are homeomorphic to $H(I)$, the hyperspace of countable compact subsets of the interval.","For dense-in-itself 0-dimensional Polish $X$, $A_n(X) \\cong Q^\\omega$ and $A_\\omega(X) \\cong S_4$; in particular, the rationals, the irrationals, and the Cantor set all give the same hyperspace types at these levels.","The $c_0$ model for $A_1(X,\\{p\\})$ says the space of compacta converging to a specified point in a Peano continuum has a universal type, independent of the continuum, once the point has order at least 2.","The paper explicitly answers open questions about whether the hyperspace of convergent sequences on the closed interval is homeomorphic to that on the open interval, and whether the corresponding hyperspaces on the circle are contractible."],"supporting_citations":[{"why":"Supplies the equivalence theorem that turns everywhere preuniversality into strong universality for dense coideals in Lawson semilattices, the load-bearing step for the absorbing-pair arguments.","marker":"[2]"},{"why":"Characterizes the hyperspace of countable compact subsets of the interval as an absorbing set, providing the target homeomorphism type $H(I)$.","marker":"[7]"},{"why":"Shows the hyperspace of a Peano continuum is a Hilbert cube, fixing the ambient manifold for the absorbing-pair computations.","marker":"[14]"},{"why":"Provides the standard complete sets for the relevant Borel classes and the completeness arguments used to locate the hyperspaces in the Borel hierarchy.","marker":"[23]"},{"why":"Establishes uniqueness of $C$-absorbing spaces, the principle that converts absorbing properties into explicit homeomorphisms.","marker":"[5]"},{"why":"Supplies the absorbing-pair theorems and pair-universality facts used throughout Sections 4 to 11.","marker":"[1]"},{"why":"Proves dense locally path-connected subsemilattices of Lawson semilattices are ANRs homotopy dense in the ambient semilattice, giving the ANR/AR structure of the hyperspaces.","marker":"[24]"},{"why":"Gives the homeomorphism criterion for zero-dimensional first-category spaces in given Borel classes used for the $Q^\\omega$ and $S_4$ characterizations.","marker":"[31]"}],"fun_headline_variants":["Descriptive class decides countable-compacta hyperspace type","Countable-compacta hyperspaces: topology determined by derived sets","Hyperspaces of countable compacta: fixed shapes for fixed classes","Countable-compacta hyperspaces classified into c0, Q^omega, and more"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a cited theorem for Lawson semilattices asserting that, for a dense coideal that is locally path-connected in the semilattice, everywhere preuniversality of the pair implies strong universality; if that implication is false or needs an extra hypothesis, the proofs of the main homeomorphisms $A_\\omega(X) \\cong \\Sigma_4$ and $H(X) \\cong H(I)$ lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Descriptive class decides countable-compacta hyperspace type","Countable-compacta hyperspaces: topology determined by derived sets","Hyperspaces of countable compacta: fixed shapes for fixed classes","Countable-compacta hyperspaces classified into c0, Q^omega, and more"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000828,"raw_usage":{"total_tokens":3717,"prompt_tokens":1145,"completion_tokens":2572,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":761,"completion_tokens_details":{"reasoning_tokens":2494}},"tokens_in":761,"tokens_out":2572,"duration_ms":20320,"temperature":1.0,"reasoning_tokens":2494,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:34:42.319663+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a Lawson semilattice $M$ and a dense coideal $X$ that is locally path-connected in $M$ for which the pair $(M,X)$ is everywhere preuniversal but not strongly universal for some $\\Pi^0_2$-hereditary class $C$; that directly refutes the equivalence theorem used in Theorem 8.3 and would invalidate the absorbing-set route to Theorem 8.5. A less surgical test: find two nondegenerate connected locally connected Polish spaces, one locally compact and one nowhere locally compact, whose hyperspaces $H(X)$ and $H(Y)$ are not homeomorphic.","supporting_citations":[{"cited_title":"Banakh, The strong universality of ANRs with a suitabl e algebraic structure, preprint","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence theorem that turns everywhere preuniversality into strong universality for dense coideals in Lawson semilattices, the load-bearing step for the absorbing-pair arguments."},{"cited_title":"Cauty, Caract´ erisation topologique de l’espace des fonctions d´ erivables, Fund","cited_arxiv_id":null,"evidence_quote":"Characterizes the hyperspace of countable compact subsets of the interval as an absorbing set, providing the target homeomorphism type $H(I)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the hyperspace of a Peano continuum is a Hilbert cube, fixing the ambient manifold for the absorbing-pair computations."},{"cited_title":"Kechris, Classical descriptive set theory, Springe r, 1995","cited_arxiv_id":null,"evidence_quote":"Provides the standard complete sets for the relevant Borel classes and the completeness arguments used to locate the hyperspaces in the Borel hierarchy."},{"cited_title":"Kubi´ s, K","cited_arxiv_id":null,"evidence_quote":"Proves dense locally path-connected subsemilattices of Lawson semilattices are ANRs homotopy dense in the ambient semilattice, giving the ANR/AR structure of the hyperspaces."},{"cited_title":"Steel, Analytic sets and Borel isomorphisms","cited_arxiv_id":null,"evidence_quote":"Gives the homeomorphism criterion for zero-dimensional first-category spaces in given Borel classes used for the $Q^\\omega$ and $S_4$ characterizations."}],"review_version":1}