Pith. sign in

REVIEW 1 major objections 6 minor 47 references

Compactly generated spaces and quasi-spaces in topology

T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read This paper proves that compactly generated spaces and quasi-spaces, two classical constructions around the category Top, both exist and are cartesian closed in the general setting of (T,V)-categories.

desk verdict Solid (T,V)-generalization of compactly generated and quasi-spaces; the Alexandroff examples are the payoff, and the flagged gap in 4.3 is real but repairable. read the letter →

arxiv 1908.04287 v1 pith:LJWNSYHF submitted 2019-08-12 math.CT math.GN

Willian Ribeiro This is my paper · ORCID
classification math.CTmath.GN MSC 18B3018D1554A0554B3054C3554D50
keywords (TV)-categoriescompactandHausdorffspacescompactlygeneratedAlexandroffcartesianclosednessquasi-spacesquasi-topologicaltopologicalfunctors
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 sets out to prove that two classical strategies for repairing the failure of cartesian closedness in the category of topological spaces—restricting to compactly generated spaces and enlarging to quasi-spaces—work at the much higher level of generality of (T,V)-categories, a framework that includes ordered, metric, topological, and approach spaces in a single formalism. It shows that for any class C of 'generating' objects satisfying a natural product condition, the C-generated (T,V)-spaces form a cartesian closed category, and that taking C to be the compact Hausdorff spaces yields the expected compactly generated case. It then defines quasi-(T,V)-spaces by axiomatizing which maps from compact Hausdorff objects are admissible, proves the resulting category is topological and cartesian closed, and shows that compactly generated spaces embed into quasi-spaces as a full reflective subcategory. The upshot is that the compactly generated / quasi-space dichotomy is not a peculiarity of Top but a structural feature of the entire (T,V)-landscape.

What carries the argument

The machinery has two load-bearing pieces. First is the notion of a generating class $\mathcal{C}$ together with 'probes': continuous maps $C\to(X,a)$ with $C\in\mathcal{C}$. The $\mathcal{C}$-generated structure $a_c$ is the final (T,V)-structure with respect to all probes, so that a space is $\mathcal{C}$-generated exactly when it is a coequalizer of a coproduct of generating spaces; the exponential structure on $\mathcal{C}$-Map$(Y,Z)$ is built from the initial structure induced by composing with probes. Second is the quasi-space structure: a set $X$ together with, for each compact Hausdorff $C$, a set $Q(C,X)$ of admissible maps closed under constant maps, precomposition, and a covering axiom (QS3) that says a map is admissible precisely when a finite family of admissible maps covers it through a surjective continuous map from a coproduct. The cartesian closedness of Qs(T,V)-Cat is carried by the exponential quasi-structure on Qs$(X,Y)$, whose verification of (QS3) uses pullbacks of covering families and the distributivity of (T,V)-Cat.

What would settle it

Within one of the paper's own examples, say App = $(\mathsf{U},P_+)$-Cat, take finite families of compact Hausdorff objects $C_i$ and $B$ for which the canonical map $\coprod(C_i\times_C B)\to(\coprod C_i)\times_C B$ can be computed; if any such map is not an isomorphism in (T,V)-Cat, the unstated distributive premise behind Section 4.3 fails, and the proof of cartesian closedness of Qs(T,V)-Cat collapses. Conversely, checking these maps in each row of the table would confirm the premise the argument silently uses.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that both classical constructions survive the passage from Top to (T,V)-Cat. Theorem 2.3 establishes that C-Map and (T,V)-Cat$_\mathcal{C}$ are cartesian closed whenever each generating object is exponentiable and products of generating objects are C-generated; Theorem 4.3 establishes that Qs(T,V)-Cat, whose objects are sets equipped with admissible-map data $Q(C,X)$ satisfying axioms (QS1)–(QS3), is also cartesian closed; and Section 5 establishes that the compactly generated (T,V)-spaces form a full reflective subcategory of Qs(T,V)-Cat, with the reflector sending a quasi-space to the final structure induced by its admissible maps. Along the way the paper recovers, in the table of examples, the classical facts that compactly generated topological spaces are quotients of disjoint sums of compact Hausdorff spaces, that Alexandroff topological spaces are exactly the C-generated spaces for the Sierpiński generator, and that quasi-spaces in the approach and $(\mathsf{U},[0,1]_\odot)$ settings coincide with quasi-topological spaces because the compact Hausdorff objects reduce to $\mathsf{Set}^{\mathsf{U}}$.

Load-bearing premise

The proof that function quasi-spaces are closed under the covering axiom assumes, without proof, that (T,V)-Cat is distributive in the specific sense that the canonical comparison $\coprod_i(C_i\times_C B)\to(\coprod_i C_i)\times_C B$ is an isomorphism; if that fails, the verification of axiom (QS3) for the exponential quasi-structure collapses, and Theorem 4.3 is not established.

Editorial extensions

If this is right

  • Every row of the paper's table—ordered, metric, ultrametric, bounded metric, topological, approach, non-Archimedean approach, and $(\mathsf{U},[0,1]_\odot)$-categories—has a cartesian closed category of compactly generated objects.
  • The Alexandroff construction generalizes: for an integral totally ordered quantale and $T=I$, the Sierpiński $(V,\mathrm{hom})$ object generates a cartesian closed category of Alexandroff V-spaces, and in Top this recovers classical Alexandroff spaces, equivalent to preordered sets.
  • Quasi-(T,V)-spaces form a topological category over Set, hence are complete and cocomplete, and they are cartesian closed with the expected exponential Qs$(X,Y)$.
  • A (T,V)-space is compactly generated exactly when, for every (T,V)-space $Y$, (T,V)-continuous maps $X\to Y$ coincide with quasi-continuous maps between the associated quasi-spaces; consequently compactly generated spaces form a full reflective subcategory of all quasi-spaces.
  • In several important cases the quasi-space category is independent of the metric choice: QsApp, QsNA-App, Qs$(\mathsf{U},[0,1]_\odot)$-Cat, and QsTop all coincide because their compact Hausdorff objects are all $\mathsf{Set}^{\mathsf{U}}$.

Reading between the lines

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

  • The paper leaves the distributive law in Section 4.3 unproved; a natural next step is to verify it for each row of the table, and the case-by-case diagram checks already used in Section 3.2 suggest the law may be forced by the algebraic extension $\xi$ whenever the quantale is integral and totally ordered.
  • Because quasi-spaces only consult the compact Hausdorff objects, the equality QsApp = QsTop suggests a general principle: the quasi-space category over a (T,V)-category depends only on the 'compact Hausdorff core' $\mathsf{Set}^{\mathsf{T}}$, so two doctrines with the same T-algebras will have identical quasi-space categories.
  • A question the paper does not address is whether the reflector from quasi-spaces to compactly generated spaces preserves finite products; if it does, the quasi-space exponential would automatically restrict to the compactly generated subcategory, giving an alternative route to Theorem 2.3.
  • For Alexandroff spaces, the two diagram conditions, commutativity of (3.ii) and inequality (3.iii), are shown to be sufficient for $A_\circ$ to give all Alexandroff (T,V)-spaces from Alexandroff V-spaces; an extension the paper does not pursue is whether these conditions are also necessary in general.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper develops a general framework of C-generated (T,V)-spaces and quasi-(T,V)-spaces, extending the classical notions of compactly generated topological spaces and Spanier quasi-topological spaces to categories of lax algebras for a monad T and a quantale V. Section 2 defines C-generated spaces, proves that (T,V)-Cat_C is coreflective in (T,V)-Cat, characterizes C-generated spaces as coequalizers of coproducts of generating spaces, and, under an explicit condition (EP), proves that both C-Map and (T,V)-Cat_C are cartesian closed. Section 3 applies the framework to the classes of compact Hausdorff spaces, the Sierpinski (T,V)-space (yielding Alexandroff spaces), exponentiable spaces, and injective spaces, with detailed verifications for App, NA-App, and (U,[0,1]⊙)-Cat. Section 4 defines quasi-(T,V)-spaces, proves that the forgetful functor Qs(T,V)-Cat -> Set is topological, and proves that Qs(T,V)-Cat is cartesian closed. Section 5 establishes that the category of compactly generated (T,V)-spaces is fully reflective in Qs(T,V)-Cat, generalizing Day's result for Top.

Significance. If the results hold, the paper provides a coherent and broad generalization of two classical remedies for the non-cartesian closedness of Top: the subcategory of compactly generated spaces and the supercategory of quasi-topological spaces. The main theorems are proven by standard initial and final lifting arguments, and the paper follows the strategy of Escardo-Lawson-Simpson and Day closely, which makes the claims concrete and falsifiable. The treatment of Alexandroff spaces and the detailed examples for approach spaces and (U,[0,1]⊙)-Cat are valuable. The explicit reliance on [CHR20] for exponentiability of injective spaces is acknowledged. The paper contains no machine-checked proofs, but the arguments are sufficiently explicit to be checked by hand; the statements are precise enough that a counterexample to any of the main theorems would be identifiable.

major comments (1)
  1. [Section 4.3, verification of (QS3) for the exponential quasi-structure] The proof of Theorem 4.3 contains the sentence "We observe that we also use distributivity of (T,V)-Cat" and describes the map µ: ∐_i(C_i ×_C B) → (∐_i C_i) ×_C B as a surjective (T,V)-continuous map. No proof or reference for this distributivity is supplied, and if the verification of (QS3) genuinely required it, Theorem 4.3 would be unsupported. On inspection, however, distributivity is not needed: the canonical map µ exists by the universal property of the coproduct, and the covering map can be taken to be π_B ∘ µ, whose surjectivity follows directly from the surjectivity of η and the pullback condition. The triangle (ev ∘ ⟨β·h, α⟩) ∘ (π_B ∘ µ) = ∐_i γ_i then holds by the definition of the pullback. I therefore recommend deleting or correcting the distributivity sentence and adding a one-line justification that π_B ∘ µ is a continuous surjection. This resolves the main proof gap without changing the statement of Theorem 4.3.
minor comments (6)
  1. [Section 2.3, formula for d_j] In the displayed definition of the exponential structure d_j, the projections are named π_X and π_Z and the evaluation is written as h(x); since the exponential is Z^{Y_j}, this should be π_{Y_j} and π_Z and h(y_j), following [CHT03].
  2. [Section 4.2, quotient quasi-structure] The verification of (QS3) for the quotient quasi-structure is relegated to "One can check"; since (QS3) is a defining axiom of a quasi-space, please include the short argument that uses the finite coproduct of the covering maps together with closure of C under finite coproducts.
  3. [Section 3.2, Example (3), NA-App] The sentence "ξ(v1)>ξ(v2)=0" is confusing after the preceding claim ξ(v1)=ξ(v2)=v; please clarify that the value of the operation ξ(v1)>ξ(v2) is 0 and that this is what makes the relevant diagram non-commutative.
  4. [Section 1.3, before equation (1.x)] The phrase "for each X∈TTX" should be typeset as an element of T^2X (or the intended iterated power of T) to avoid ambiguity.
  5. [Section 1.3, Theorem statements and diagram (1.ix)] In Theorem 1.3 the first hypothesis says diagram (1.ix) is commutative, but (1.ix) is displayed with a lax-commutativity symbol; please specify whether strict commutativity is intended and adjust the diagram accordingly.
  6. [Section 4.3, Examples] The equalities QsApp = QsNA-App = QsTop and QsOrd = QsMet = QsUltMet = QsB1Met are asserted from the equivalences of the compact Hausdorff classes; a brief justification that the quasi-space definitions are preserved under these equivalences would be helpful for the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are conditional derivations from prior independent results; the only notable weakness is an unproved distributivity assertion in the proof of Theorem 4.3, which is a proof gap rather than a circular reduction.

full rationale

The paper's derivation chain is not circular. Theorem 2.3 proves cartesian closedness of C-generated (T,V)-spaces under condition (EP) by the standard ELS04 transpose argument: the internal hom structure on C-Map(Y,Z) is initially lifted from exponentials Z^{Y_j} along probes, and the bijection f C-continuous iff its transpose is C-continuous is verified directly from the definitions and (EP). For the compact Hausdorff class, (EP) is imported from [HST14] and [CHR20] with stated hypotheses (T-algebra structure, condition (1.x), closure under products/coproducts) that do not include the target conclusion; no fitted parameter is later renamed as a prediction. Theorem 4.3 constructs the quasi-space exponential by defining admissibility on Qs(X,Y) precisely so that evaluation is quasi-continuous, then proves the transpose is quasi-continuous from that defining condition; this is the usual construction of an internal hom, not a restatement of cartesian closedness. Section 5's full reflection result is a direct verification using the final lifting of admissible maps and the characterization (5.i); it does not presuppose the reflection it proves. The self-citation [CHR20] is co-authored by the paper's author, but it is used as an external general criterion for exponentiability of injective spaces, not as a uniqueness theorem invoked to forbid alternatives, and the present paper's main constructions do not reduce to that citation. The genuinely weak point is in the proof of (QS3) in Section 4.3: after introducing the map mu, the text says 'We observe that we also use distributivity of (T,V)-Cat' and gives neither proof nor citation that the canonical comparison coproduct_i(C_i times_C B) -> (coproduct_i C_i) times_C B is a surjective continuous map. That comparison is load-bearing for the covering argument establishing admissibility of ev o <beta.h, alpha>. If distributivity fails, Theorem 4.3 is unsupported; however, this is an omitted hypothesis or proof gap, not a circular step, because distributivity is not the conclusion of Theorem 4.3 and is not derived from it. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper is a pure mathematical generalization; it introduces no free parameters or invented empirical entities. The central theorems rest on the (T,V)-category axioms, the cardinality and choice conventions, and several explicit conditions ((EP), distributivity, diagram commutativity) that are assumed or checked case by case.

assumptions (8)
  • domain assumption V is an integral, completely distributive quantale that is also a Heyting algebra.
    The entire (T,V)-setting assumes these properties of V (Section 1.2); integrality is derived from the constant-map assumption in Lemma 1.2, complete distributivity is assumed for the examples such as 2, P+, Pmax, and [0,1]⊙.
  • domain assumption T is a monad on Set satisfying the Beck-Chevalley condition, and its lax extension to V-Rel is flat and commutes with involution.
    Needed to define the lax monad on V-Rel and the topological functor |-|: (T,V)-Cat → Set (Section 1.2).
  • domain assumption Every constant map between (T,V)-spaces is continuous (equivalently k=⊤ and T1=1).
    This 'fairly restrictive condition' (Section 1.2) is assumed so that (T,V)-Cat is a topological category in the classical sense; the paper proves the equivalence but not the condition itself.
  • domain assumption V is lean.
    Used in Section 1.4 to identify compact Hausdorff (T,V)-spaces with T-algebras (Eq. 1.xiii) and thus obtain exponentiability and the equivalence (T,V)-Cat_CompHaus ≅ Set^T.
  • ad hoc to paper Condition (EP): each element of the generating class C is exponentiable in (T,V)-Cat and the product of two elements of C is a C-generated space.
    This is the key hypothesis for Theorem 2.3 (cartesian closedness of C-Map and (T,V)-Cat_C). It is assumed, not proven; the paper verifies it for compact Hausdorff spaces and for the Sierpiński space under additional conditions.
  • ad hoc to paper Distributivity of (T,V)-Cat: the canonical map ∐_i(C_i ×_C B) → (∐_i C_i) ×_C B is an isomorphism.
    Invoked without proof or citation in Section 4.3 to verify axiom (QS3) for the exponential quasi-structure; the proof of cartesian closedness of Qs(T,V)-Cat depends on it.
  • ad hoc to paper For the Alexandroff results, diagram (3.ii) is commutative and inequality (3.iii) holds.
    These conditions are hypotheses of Lemma 2 and Corollary 3.2; they are checked for App and (U,[0,1]⊙)-Cat but fail for NA-App, which is explicitly noted.
  • standard math Axiom of Choice.
    Granted explicitly in Section 4.1(I) to conclude that the quasi-space associated with a (T,V)-space satisfies (QS3).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Compactly generated spaces and quasi-spaces in topology." pith.science (2026). https://pith.science/paper/LJWNSYHF

@misc{pith2026190804287,
  author       = {Pith},
  title        = {Pith review of: Compactly generated spaces and quasi-spaces in topology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJWNSYHF}},
  note         = {Machine review of arXiv:1908.04287}
}
abstract

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $\mathsf{Top}$ of topological spaces and continuous functions, to study $\textit{compactly generated spaces}$ and $\textit{quasi-spaces}$ in this setting. Moreover, for a class $\mathcal{C}$ of objects we generalize the notion of $\textit{$\mathcal{C}$-generated spaces}$, from which we derive, for instance, a general concept of $\textit{Alexandroff spaces}$. Furthermore, as done for $\mathsf{Top}$, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Strecker

    Ji r \'i Ad\'amek, Horst Herrlich, and George E. Strecker. Abstract and concrete categories . Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, 1990. The joy of cats, A Wiley-Interscience Publication

    work page 1990

  2. [2]

    Peter I. Booth. A unified treatment of some basic problems in homotopy theory. Bull. Amer. Math. Soc. , 79:331--336, 1973

    work page 1973

  3. [3]

    Coproducts and ultrafilters

    Reinhard B\" o rger. Coproducts and ultrafilters. J. Pure Appl. Algebra , 46(1):35--47, 1987

    work page 1987

  4. [4]

    Sarah L. Browne. E-theory spectra for graded C *-algebras. arXiv:1708.03258, 2017

    work page Pith review arXiv 2017

  5. [5]

    Representable ( T , V ) -categories

    Dimitri Chikhladze, Maria Manuel Clementino, and Dirk Hofmann. Representable ( T , V ) -categories. Appl. Categ. Structures , 23(6):829--858, 2015

    work page 2015

  6. [6]

    Topological features of lax algebras

    Maria Manuel Clementino and Dirk Hofmann. Topological features of lax algebras. Appl. Categ. Structures , 11(3):267--286, 2003

    work page 2003

  7. [7]

    Effective descent morphisms in categories of lax algebras

    Maria Manuel Clementino and Dirk Hofmann. Effective descent morphisms in categories of lax algebras. Appl. Categ. Structures , 12(5-6):413--425, 2004

    work page 2004

  8. [8]

    On extensions of lax monads

    Maria Manuel Clementino and Dirk Hofmann. On extensions of lax monads. Theory Appl. Categ. , 13:No. 3, 41--60, 2004

    work page 2004

Show all 47 references
  1. [9]

    Exponentiation in V -categories

    Maria Manuel Clementino and Dirk Hofmann. Exponentiation in V -categories. Topology Appl. , 153(16):3113--3128, 2006

    2006

  2. [10]

    Lawvere completeness in topology

    Maria Manuel Clementino and Dirk Hofmann. Lawvere completeness in topology. Appl. Categ. Structures , 17(2):175--210, 2009

    2009

  3. [11]

    The monads of classical algebra are seldom weakly C artesian

    Maria Manuel Clementino, Dirk Hofmann, and George Janelidze. The monads of classical algebra are seldom weakly C artesian. J. Homotopy Relat. Struct. , 9(1):175--197, 2014

    2014

  4. [12]

    Cartesian closed exact completions in topology

    Maria Manuel Clementino, Dirk Hofmann, and Willian Ribeiro. Cartesian closed exact completions in topology. J. Pure Appl. Algebra , 224(2):610--629, 2020

    2020

  5. [13]

    Exponentiability in categories of lax algebras

    Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen. Exponentiability in categories of lax algebras. Theory Appl. Categ. , 11(15):337--352, 2003

    2003

  6. [14]

    Metric, topology and multicategory---a common approach

    Maria Manuel Clementino and Walter Tholen. Metric, topology and multicategory---a common approach. J. Pure Appl. Algebra , 179(1-2):13--47, 2003

    2003

  7. [15]

    From lax monad extensions to topological theories

    Maria Manuel Clementino and Walter Tholen. From lax monad extensions to topological theories. Categorical methods in algebra and topology , Textos Mat./Math. Texts 46:99--123, Univ. Coimbra, 2014

    2014

  8. [16]

    Topological properties of non- A rchimedean approach spaces

    Eva Colebunders and Karen Van Opdenbosch. Topological properties of non- A rchimedean approach spaces. Theory Appl. Categ. , 32:Paper No. 41, 1454--1484, 2017

    2017

  9. [17]

    Relationship of Spanier's quasitopological spaces to k-spaces

    Brian Day. Relationship of Spanier's quasitopological spaces to k-spaces. M.Sc. Thesis, Univ. of Sydney, 1968

    1968

  10. [18]

    A reflection theorem for closed categories

    Brian Day. A reflection theorem for closed categories. J. Pure Appl. Algebra , 2(1):1--11, 1972

    1972

  11. [19]

    Free finitary algebras on compactly generated spaces

    Brian Day. Free finitary algebras on compactly generated spaces. Bull. Austral. Math. Soc. , 24(2):277--288, 1981

    1981

  12. [20]

    Dubuc and Luis Espa \ n ol

    Eduardo J. Dubuc and Luis Espa \ n ol. Quasitopoi over a base category. arXiv:math/0612727v1 [math.CT], 2006

    2006 arXiv

  13. [21]

    E-theory for C ^* -algebras over topological spaces

    Marius Dadarlat and Ralf Meyer. E-theory for C ^* -algebras over topological spaces. J. Funct. Anal. , 263(1):216--247, 2012

    2012

  14. [22]

    Dubuc and Horacio Porta

    Eduardo J. Dubuc and Horacio Porta. Uniform spaces, S panier quasitopologies, and a duality for locally convex algebras. J. Austral. Math. Soc. Ser. A , 29(1):99--128, 1980

    1980

  15. [23]

    Eduardo J. Dubuc. Concrete quasitopoi. In Applications of sheaves ( P roc. R es. S ympos. A ppl. S heaf T heory to L ogic, A lgebra and A nal., U niv. D urham, D urham, 1977) , volume 753 of Lecture Notes in Math. , pages 239--254. Springer, Berlin, 1979

    1977

  16. [24]

    Topologies on spaces of continuous functions

    Mart\'in Escard\'o and Reinhold Heckmann. Topologies on spaces of continuous functions. Topology Proc. , 26(2):545--564, 2001

    2001

  17. [25]

    Comparing C artesian closed categories of (core) compactly generated spaces

    Mart\'in Escard\'o, Jimmie Lawson, and Alex Simpson. Comparing C artesian closed categories of (core) compactly generated spaces. Topology Appl. , 143(1-3):105--145, 2004

    2004

  18. [26]

    Compact sets of functions and function rings

    David Gale. Compact sets of functions and function rings. Proc. Amer. Math. Soc. , 1:303--308, 1950

    1950

  19. [27]

    Cartesian closed topological categories

    Horst Herrlich. Cartesian closed topological categories. Math. Colloq. Univ. Cape Town , 9:1--16, 1974

    1974

  20. [28]

    An algebraic description of regular epimorphisms in topology

    Dirk Hofmann. An algebraic description of regular epimorphisms in topology. J. Pure Appl. Algebra , 199(1-3):71--86, 2005

    2005

  21. [29]

    Topological theories and closed objects

    Dirk Hofmann. Topological theories and closed objects. Adv. Math. , 215(2):789--824, 2007

    2007

  22. [30]

    Injective spaces via adjunction

    Dirk Hofmann. Injective spaces via adjunction. J. Pure Appl. Algebra , 215(3):283--302, 2011

    2011

  23. [31]

    The enriched V ietoris monad on representable spaces

    Dirk Hofmann. The enriched V ietoris monad on representable spaces. J. Pure Appl. Algebra , 218(12):2274--2318, 2014

    2014

  24. [32]

    The quasicategory of quasispaces is illegitimate

    Horst Herrlich and Minakshisundaram Rajagopalan. The quasicategory of quasispaces is illegitimate. Arch. Math. (Basel) , 40(4):364--366, 1983

    1983

  25. [33]

    Dirk Hofmann and Carla D. Reis. Probabilistic metric spaces as enriched categories. Fuzzy Sets and Systems , 210:1--21, 2013

    2013

  26. [34]

    Dirk Hofmann and Gavin J. Seal. Exponentiable approach spaces. Houston J. Math. , 41(3):1051--1062, 2015

    2015

  27. [35]

    Seal, and Walter Tholen, editors

    Dirk Hofmann, Gavin J. Seal, and Walter Tholen, editors. Monoidal topology. A categorical approach to order, metric, and topology , volume 153 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, July 2014. Authors: Maria Manuel Clementi...

    2014

  28. [36]

    Lawvere completion and separation via closure

    Dirk Hofmann and Walter Tholen. Lawvere completion and separation via closure. Appl. Categ. Structures , 18(3):259--287, 2010

    2010

  29. [37]

    John L. Kelley. General topology . D. Van Nostrand Company, Inc., Toronto-New York-London, 1955

    1955

  30. [38]

    William Lawvere

    F. William Lawvere. Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matem\`atico e Fisico di Milano, 43(1):135--166, December 1973. Republished in: Reprints in Theory and Applications of Categories, No. 1 (2002), 1--37

    2002

  31. [39]

    Approach spaces

    Robert Lowen. Approach spaces . Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1997. The missing link in the topology-uniformity-metric triad, Oxford Science Publications

    1997

  32. [40]

    Categories for the working mathematician

    Saunders Mac Lane. Categories for the working mathematician . Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971

    1971

  33. [41]

    Ernest G. Manes. Compact H ausdorff objects. General Topology and Appl. , 4:341--360, 1974

    1974

  34. [42]

    Louis D. Nel. Cartesian closed coreflective hulls. Quaestiones Math. , 2(1-3):269--283, 1977/78

    1977

  35. [43]

    Constructive Topology of Bishop Spaces

    Iosif Petrakis. Constructive Topology of Bishop Spaces . PhD thesis, Ludwig-Maximilians-University of Munich, 2015

    2015

  36. [44]

    Quasi-topologies

    Edwin Spanier. Quasi-topologies. Duke Math. J. , 30:1--14, 1963

    1963

  37. [45]

    Steenrod

    Norman E. Steenrod. A convenient category of topological spaces. Michigan Math. J. , 14:133--152, 1967

    1967

  38. [46]

    Rainer M. Vogt. Convenient categories of topological spaces for homotopy theory. Arch. Math. (Basel) , 22:545--555, 1971

    1971

  39. [47]

    A constructive model of uniform continuity

    Chuangjie Xu and Mart\' i n Escard\' o . A constructive model of uniform continuity. In Typed lambda calculi and applications , volume 7941 of Lecture Notes in Comput. Sci. , pages 236--249. Springer, Heidelberg, 2013

    2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.