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arxiv: 2606.27943 · v1 · pith:6OMCJ7LSnew · submitted 2026-06-26 · 🧮 math.GN

New approaches to remote points

Serhii Bardyla , Jaroslav \v{S}upina This is my paper

Pith reviewed 2026-06-29 02:05 UTC · model grok-4.3

classification 🧮 math.GN
keywords remote pointsStone-Čech compactificationnowhere dense setsω-bounded spacesBoolean algebrasTychonoff spacestopological ultrafilters
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The pith

Remote points in the Stone-Čech compactification of a Tychonoff space correspond to ultrafilters on the Boolean algebra of open and nowhere dense sets, and those of the real line form an ω-bounded space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a remote point p in βX minus X as one that avoids the closure of every nowhere dense subset of X. It gives characterizations of the existence of such points through topological ultrafilters, measures, and compact-like properties linked to the nowhere dense ideal. The remote points space is shown to be homeomorphic to a subspace of the Stone space of the smallest Boolean algebra that contains both the open sets and the nowhere dense sets. For the real line specifically, this space of remote points is proved to be ω-bounded.

Core claim

A point p in βX ∖ X is remote when it lies outside the closure of every nowhere dense subset of the Tychonoff space X. The space of all remote points is homeomorphic to a subspace of the Stone space taken over the smallest Boolean algebra containing all open sets and all nowhere dense sets. When X is the real line, the space of its remote points is ω-bounded.

What carries the argument

The homeomorphism that identifies the space of remote points with a subspace of the Stone space of the Boolean algebra generated by open sets and nowhere dense sets.

If this is right

  • The existence of remote points in βX ∖ X can be decided by the presence of ultrafilters on the relevant Boolean algebra.
  • The remote points of the real line form an ω-bounded space.
  • Remote points admit equivalent descriptions via measures and compact-like properties relative to the nowhere dense ideal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Boolean-algebra approach may allow similar embeddings for points defined by other ideals in compactifications.
  • ω-boundedness of the remote points of ℝ could be used to study selection principles or tightness in the remainder.
  • The construction might extend to characterize remote points in other compactifications beyond the Stone-Čech one.

Load-bearing premise

The characterizations and homeomorphism are derived under the assumption that X is Tychonoff and the constructions use the ideal of nowhere dense subsets.

What would settle it

An explicit Tychonoff space X where the remote points fail to be homeomorphic to any subspace of the described Stone space, or a countable subset of remote points in βℝ ∖ ℝ whose closure is not compact.

read the original abstract

For a given Tychonoff space $X$, a point $p\in \beta(X)\setminus X$ is called {\em remote} if $p$ is not in the closure of any nowhere dense subset of $X$. In this paper, we characterize spaces with remote points in terms of certain topological ultrafilters, measures, and compact-like properties corresponding to the ideal consisting of nowhere dense sets. It is shown that the space of remote points is homeomorphic to a subspace of the Stone space taken over the smallest Boolean algebra containing all open and nowhere dense sets. Also, we show that the space of remote points of $\mathbb R$ is $\omega$-bounded.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper characterizes remote points p in βX \ X (for Tychonoff X) as those not in the closure of any nowhere dense subset of X. It claims characterizations of spaces possessing remote points via topological ultrafilters, measures, and compact-like properties relative to the nowhere dense ideal; proves that the space of remote points is homeomorphic to a subspace of the Stone space of the smallest Boolean algebra containing all open sets and nowhere dense sets; and shows that the remote points of ℝ form an ω-bounded space.

Significance. If the claimed characterizations and homeomorphism hold, the results supply new links between remote points, Boolean-algebra constructions, and ideal-theoretic properties, which may prove useful for studying remainders in Stone-Čech compactifications. The concrete ω-boundedness statement for ℝ is a verifiable application that could serve as a test case for the general framework.

minor comments (1)
  1. The abstract states the main theorems but supplies no proofs, derivations, or section references, making it impossible to verify whether the characterizations follow from the given definitions of remote points and the nowhere dense ideal.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript, which accurately reflects the abstract and main results on characterizations of remote points via ultrafilters on the nowhere dense ideal, the homeomorphism to a subspace of the Stone space, and the ω-boundedness of the remote points of ℝ. No specific major comments were provided in the report, so we have no individual points to address. We would welcome any additional feedback to address the uncertainty in the recommendation.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's core claims rest on direct constructions: remote points are defined as points in βX ∖ X outside closures of nowhere dense sets; the space is shown homeomorphic to a subspace of the Stone space of the Boolean algebra generated by open sets and nowhere dense sets; and the ℝ case is shown ω-bounded. These are standard applications of Stone-Čech compactification and Boolean algebra theory for Tychonoff spaces with respect to the nowhere dense ideal. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise depends on a self-citation chain. The derivation chain is self-contained against external topological benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper operates in standard ZFC set theory and general topology; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • domain assumption X is a Tychonoff space
    Explicitly stated as the setting for the definition of remote points in the first sentence of the abstract.
  • standard math Standard properties of the Stone-Čech compactification βX and the ideal of nowhere dense sets
    Invoked throughout the abstract in the definition of remote points and the constructions involving ultrafilters and Boolean algebras.

pith-pipeline@v0.9.1-grok · 5634 in / 1335 out tokens · 43602 ms · 2026-06-29T02:05:41.366950+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

32 extracted references

  1. [1]

    O. Alas, L. Junqueira, R. Wilson,The degree of weakly discretely generated spaces, Acta Math. Hung. 143(2014), 453—465

    2014

  2. [2]

    Bardyla, J

    S. Bardyla, J. ˇSupina, L. Zdomskyy,Open filters and measurable cardinals, Arch. Math. Logic65 (2026), 41–70. 16

    2026

  3. [3]

    Borodulin–Nadzieja, A

    P. Borodulin–Nadzieja, A. Ranchynski,On points avoiding measures, Topology Appl.354(2024), 108988

    2024

  4. [4]

    Borodulin–Nadzieja, G

    P. Borodulin–Nadzieja, G. Plebanek,Measures on Suslinean spaces, Fund. Math.235(2016), 287–302

    2016

  5. [5]

    Brown,Remote points in products under CH, Topology Appl.155(2008), 190-200

    J. Brown,Remote points in products under CH, Topology Appl.155(2008), 190-200

    2008

  6. [6]

    Brown, A

    J. Brown, A. Dow,Remote points under the continuum hypothesis, Algebra Univers.60(2009), 345– 364

    2009

  7. [7]

    Bukovsk´ y, The structure of the real line, Basel Boston, Birkha¨ user 2011

    L. Bukovsk´ y, The structure of the real line, Basel Boston, Birkha¨ user 2011

    2011

  8. [8]

    Carlson,Balanced Z-filters, Top

    J. Carlson,Balanced Z-filters, Top. Proc.10(1985), 17–32

    1985

  9. [9]

    Carlson,Prime Wallman compactification, Top

    J. Carlson,Prime Wallman compactification, Top. Proc.12(1987), 217–238

    1987

  10. [10]

    Carlson,Balanced filters, Top

    J. Carlson,Balanced filters, Top. Proc.14(1989), 25–40

    1989

  11. [11]

    Comfort, S

    W. Comfort, S. Negrepontis,The theory of ultrafilters, Springer-Verlag Berlin, Heidelberg, 1974

    1974

  12. [12]

    S. Chae, J. Smith,Remote points and G-spaces, Topology Appl.11(1980), 243–246

    1980

  13. [13]

    van Douwen,P-remote points ofX, Topology Appl

    E. van Douwen,P-remote points ofX, Topology Appl. 51 (1993), 141–146

    1993

  14. [14]

    van Douwen,Remote points, Dissertationes Math.188(1981), 1–45

    E. van Douwen,Remote points, Dissertationes Math.188(1981), 1–45

    1981

  15. [15]

    van Douwen, J

    E. van Douwen, J. van Mill,Spaces without remote points, Pacific J. Math.105(1983), 69–75

    1983

  16. [16]

    Dow,Remote points in spaces withπ-weightω 1, Fund

    A. Dow,Remote points in spaces withπ-weightω 1, Fund. Math.124(1984), 197–205

    1984

  17. [17]

    Dow,A separable space with no remote points, Trans

    A. Dow,A separable space with no remote points, Trans. Amer. Math. Soc.312(1989), 335–353

    1989

  18. [18]

    A. Dow, R. Hern´ andez-Guti´ errez,Far points and discretely generated spaces, Colloq. Math.158 (2019), 233–254

    2019

  19. [19]

    A. Dow, T. Peters,Game strategies yield remote points, Topology Appl.27:3 (1987), 245–256

    1987

  20. [20]

    A. Dow, T. Peters,Products and remote points: examples and counterexamples, Proc. Amer. Math. Soc.104(1988), 1296—1304

    1988

  21. [21]

    Drygier, G

    P. Drygier, G. Plebanek,Nonseparable growth of the integers supporting a measure, Topology Appl. 191(2015), 58–64

    2015

  22. [22]

    Dube,Remote points and the like in pointfree topology, Acta Math

    T. Dube,Remote points and the like in pointfree topology, Acta Math. Hung.123(2009), 203–222

    2009

  23. [23]

    Dˇ zamonja, G

    M. Dˇ zamonja, G. Plebanek,On Efimov spaces and Radon measures, Topology Appl.154(2007), 2063–2072

    2007

  24. [24]

    Engelking,General topology, Revised and completed ed., Berlin, Heldermann, 1989

    R. Engelking,General topology, Revised and completed ed., Berlin, Heldermann, 1989

    1989

  25. [25]

    N. Fine, L. Gillman,Remote points inβR, Proc. Amer. Math. Soc. 13 (1962) 29–36

    1962

  26. [26]

    Frol´ ık,Prime filters with CIP, Comment

    Z. Frol´ ık,Prime filters with CIP, Comment. Math. Univ. Carolin.13:3 (1972), 553–575

    1972

  27. [27]

    Hern´ andez-Guti´ errez,Remote filters and discretely generated spaces, Matematicki Vesnik67(2015), 39–51

    R. Hern´ andez-Guti´ errez,Remote filters and discretely generated spaces, Matematicki Vesnik67(2015), 39–51

    2015

  28. [28]

    Hern´ andez-Guti´ errez, M

    R. Hern´ andez-Guti´ errez, M. Hruˇ s´ ak, A. Tamariz-Mascar´ ua,Spaces of remote points, Topology Appl. 159(2012), 3002–3011

    2012

  29. [29]

    Koppelberg:Topological Dualityin: J.D

    S. Koppelberg:Topological Dualityin: J.D. Monk, R. Bonnet, S. Koppelberg (eds) Handbook of Boolean Algebras. North-Holland (1989), 95–126

    1989

  30. [30]

    Mooney,Spaces with unique Hausdorff extensions, Topology Appl.61(1995), 241–256

    D. Mooney,Spaces with unique Hausdorff extensions, Topology Appl.61(1995), 241–256

    1995

  31. [31]

    Mooney,On the equivalence of the Fomin extension and the Banaschewski-Fomin- ˇSanin extension, Houston J

    D. Mooney,On the equivalence of the Fomin extension and the Banaschewski-Fomin- ˇSanin extension, Houston J. Math.24:4 (1998), 689–697

    1998

  32. [32]

    Veksler,Maximal Nowhere Dense Sets and their Applications to Problems of Existence of Remote Points and of Weak P-Points, Math

    A. Veksler,Maximal Nowhere Dense Sets and their Applications to Problems of Existence of Remote Points and of Weak P-Points, Math. Nachr.160(1991) 263–276. University of Vienna, Institute of Mathematics, Kolingasse 14-16, 1090 Vienna, Austria. Email address:sbardyla@gmail.com Institute of Mathematics, P.J. ˇSaf´arik University in Ko ˇsice, Jesenn´a 5, 040...

    1991