Non-Approachable Points in Mathematics

• Non-Approachable Points are an umbrella concept characterized by domain-specific obstructions to standard approximation in topology, set theory, and geometry.
• They illustrate failures of canonical access through completions, club systems, geodesic paths, or computability, linking local conditions to global structures.
• Their study spans diverse contexts—from metric geometry and continuum theory to optimization—offering insights into both theoretical barriers and computational hardness.

Searching arXiv for recent and relevant papers on “approachability” and related notions across topology, set theory, metric geometry, and computation. Non-approachable points are not governed by a single uniform definition across current mathematics. In the cited literature, the phrase and its near-synonyms designate several domain-specific obstructions: finite nonstandard points that cannot be approximated by standard points, ordinals outside an approachability ideal, points excluded from dense continuumwise connected traces, points of maximum inaccessibility in convex bodies, regular points through which no closed geodesic passes, fixed points inaccessible from the complement of a hedgehog, interpolation data where a rational interpolant does not exist, and critical points that cannot be approximated efficiently (Kanovei et al., 2020, Jakob, 4 Aug 2025, Anderson, 2020, Calvo et al., 2010, Adelstein et al., 2020, Biswas, 2010, Teresa et al., 2017, Ahmadi et al., 29 Jan 2026). This suggests that “non-approachability” is best treated as an umbrella notion whose exact meaning is determined by the ambient category and by the admissible mode of access.

1. Conceptual scope and recurring patterns

Several recurrent structures organize the subject. In nonstandard metric geometry, approachability concerns whether a finite point of ${}^*M$ can be approximated arbitrarily well by standard points of $M$. In set theory, approachability is encoded by club-guessing style sequences and the ideal $I[\lambda^+]$. In continuum theory, the nearest surrogates are coastal points, non-block points, and membership in proper dense semicontinua. In planar geometry, the relevant quantity is an inaccessibility functional defined by shortest boundary-to-boundary chords. On translation surfaces, the exceptional points are oblivious points, meaning regular points through which no simple closed geodesic passes. In constructive and effective settings, the phenomenon appears as sets with no points, continua without computable points, or interpolation data that are unattainable (Lubarsky, 2015, Kihara, 2011, Teresa et al., 2017, Adelstein et al., 2020).

A common pattern is that non-approachability is seldom merely local. The cited work repeatedly ties it to global structure: properness versus Heine–Borel behavior in metric completion theory, stationary-set structure at successors of singular cardinals, composant geometry in indecomposable continua, blocking sets and branched covers on translation surfaces, or complexity-theoretic hardness in nonconvex optimization. A plausible implication is that “approach” is usually mediated by a large-scale organizing object—completion, club system, dense semicontinuum, geodesic cylinder, or algorithmic search regime—rather than by an isolated pointwise condition.

2. Metric and nonstandard approachability

For a metric space $(M,d)$, the nonstandard hull is defined from the finite points

$$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$

modulo infinitesimal nearness. A finite point $x\in {}^*M$ is approachable iff

$$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$

and inapproachable otherwise. The metric completion $\overline M$ is exactly the approachable part of the full nonstandard hull $\widehat M$, and the central criterion is

$$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$

Thus finite inapproachable points are precisely the obstruction to identifying the nonstandard hull with the metric completion (Kanovei et al., 2020).

The model example is the metric universal cover of the punctured plane with coordinates $M$0, $M$1, $M$2, and metric

$M$3

In that space, points of the form $M$4 with $M$5 infinite are finite in the nonstandard sense but are not approachable from $M$6. Consequently $M$7. The same paper identifies the geometric source of the obstruction: the completion fails the Heine–Borel property and is not locally compact near the filled-in origin. Here non-approachability is not a deficiency of completeness alone; it is a bounded noncompactness phenomenon detected by finite nonstandard points (Kanovei et al., 2020).

3. Set-theoretic approachability and internal approachability

In the set-theoretic literature, approachability is formalized via Shelah’s ideal $M$8. A set $M$9 lies in $I[\lambda^+]$0 iff there are a club $I[\lambda^+]$1 and a sequence $I[\lambda^+]$2 with each $I[\lambda^+]$3 such that whenever $I[\lambda^+]$4, there is an unbounded $I[\lambda^+]$5 with

$I[\lambda^+]$6

Non-approachable points are ordinals that fail this property. At successors of singulars, the theory is often analyzed through normal subadditive colorings and $I[\lambda^+]$7-approachability (Jakob et al., 24 Mar 2025).

A major recent result shows that, assuming $I[\lambda^+]$8 and an increasing $I[\lambda^+]$9-sequence of supercompact cardinals, for every $(M,d)$0 there is a forcing extension in which there are stationarily many

$(M,d)$1

that are good but not approachable. In particular,

$(M,d)$2

for each prescribed $(M,d)$3. This separates goodness from approachability stationarily often, including at cofinality $(M,d)$4 (Jakob et al., 24 Mar 2025).

An internal-model analogue replaces ordinals by elementary submodels. For regular $(M,d)$5, a set $(M,d)$6 is internally approachable if there is a continuous sequence $(M,d)$7 of elements of $(M,d)$8 such that $(M,d)$9 and $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$0 for all $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$1. It is internally club if $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$2 contains a club in $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$3. From countably many Mahlo cardinals, one can force a model in which, for all positive $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$4 and all $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$5, there is a stationary subset of $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$6 consisting of sets that are internally club but not internally approachable. This produces failure of approachability simultaneously on infinitely many cardinals (Jakob et al., 2024).

The strongest result in this direction gives a model of $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$7, relative to class many supercompact cardinals, such that for every singular cardinal $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$8 of countable cofinality and every regular uncountable $$\operatorname{fin}({}^*M)=\{x\in {}^*M : {}^*d(x,p)<\infty \text{ for some/every } p\in M\},$$9, there are stationarily many non-approachable points of cofinality $x\in {}^*M$0 in $x\in {}^*M$1. Equivalently,

$x\in {}^*M$2

for all such $x\in {}^*M$3 and $x\in {}^*M$4. This is “total failure of approachability” at successors of singulars of countable cofinality (Jakob, 4 Aug 2025).

4. Continuum-theoretic surrogates

In nonmetric continuum theory, the relevant language is not “approachability” but coastal points, non-block points, and proper dense semicontinua. A continuum is a compact connected Hausdorff space. A semicontinuum is a continuumwise connected space, and a proper dense semicontinuum is a semicontinuum $x\in {}^*M$5 with $x\in {}^*M$6 and $x\in {}^*M$7. For $x\in {}^*M$8, if $x\in {}^*M$9 denotes the continuum component of $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$0 in $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$1, then $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$2 is coastal if $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$3 is dense for some $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$4, and $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$5 is a non-block point if $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$6 is dense for some $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$7. The paper proves the equivalence

$$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$8

Accordingly, points that are not coastal or are block points function as the continuum-theoretic analogues of non-approachable points (Anderson, 2020).

The sharpest analysis occurs in $$(\forall \varepsilon\in \mathbb{R}_{>0})(\exists x_\varepsilon\in M)\; {}^*d(x,x_\varepsilon)<\varepsilon,$$9. For a composant $\overline M$0 and its corresponding near-coherence class $\overline M$1, the paper proves the equivalence of five statements, including: $\overline M$2

$\overline M$3

$\overline M$4

If the corresponding near-coherence class contains a $\overline M$5-point, then every countable subset of the composant is a proper non-block set and a proper coastal set. If it contains no $\overline M$6-point, then the composant has no proper coastal points and no proper non-block points, and every proper semicontinuum is nowhere dense. Globally,

$\overline M$7

The same paper records a broad empirical dichotomy: every known continuum has either a proper dense semicontinuum at every point or at no points (Anderson, 2020).

5. Geometric and dynamical access obstructions

For a bounded domain $\overline M$8, the geometric notion closest to non-approachability is inaccessibility. If $\overline M$9 and $\widehat M$0, let $\widehat M$1 be the length of the chord of $\widehat M$2 through $\widehat M$3 in direction $\widehat M$4, and define

$\widehat M$5

The points of maximum inaccessibility are those where $\widehat M$6 achieves its maximum. For strictly convex bounded domains, the maximizing set $\widehat M$7 is either a point or a segment. For a convex polygon with no parallel sides, $\widehat M$8 is a point, and for a triangle this point is unique but is not any of the classical notable points listed in the paper. Here non-approachability is measured by the shortest straight boundary-to-boundary access through the point (Calvo et al., 2010).

On translation surfaces, the exceptional points are oblivious points: regular points through which there are no simple closed geodesics. The set of oblivious points on any given translation surface is finite, but there are translation surfaces with exactly $\widehat M$9 oblivious points for every $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$0, and there is a translation surface in every genus $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$1 with an oblivious point. The paper’s branched-cover constructions turn finite blocking sets into cone points, so that any closed geodesic through a lift of the blocked point would be forced through a singularity. In this setting, non-approachability means “unreachable by closed geodesics” rather than inaccessible by all geodesics (Adelstein et al., 2020).

In complex dynamics, the strongest literal inaccessibility result is realized for hedgehogs of irrationally indifferent germs. There exists a nonlinearizable germ with a hedgehog $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$2 such that the fixed point is inaccessible from $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$3. In the lifted formulation, if

$$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$4

then there is no continuous curve

$$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$5

with $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$6. The construction uses inverse renormalization to create infinitely many nested tube-like barriers in the complement, forcing every potential access path back through bounded-height bottlenecks (Biswas, 2010).

6. Constructive, effective, and algebraic absence of points

Constructive models supply a more radical version of non-approachability: geometry without globally obtainable points. One example yields a family of sets each of which is not empty in a weak constructive sense, but for which it is false that all are inhabited uniformly. A second example defines a fixed set $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$7 with no points at all: $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$8 Nevertheless the model still carries a non-trivial distance function to $$\big[\text{every finite }x\in {}^*M \text{ is approachable}\big] \iff \big[\overline M \text{ is Heine\text{-}Borel}\big] \iff \big[\widehat M=\overline M\big].$$9 and a normable Riesz space acting like a space of real-valued functions. The paper distinguishes distance from quasi-distance by replacing infima with greatest lower bounds,

$M$00

and thereby exhibits a form of geometric access that survives the absence of points (Lubarsky, 2015).

Effective planar topology shows that even contractibility does not restore computable access. There exists a contractible planar $M$01 dendroid without computable points, solving a question of Le Roux and Ziegler negatively. The paper also proves several global incomputability phenomena: there is a computable dendrite which does not $M$02-include a co-c.e. tree; there is a co-c.e. dendrite which does not $M$03-include a computable dendrite; and there is a computable dendroid which does not $M$04-include a co-c.e. dendrite. Here $M$05-inclusion means arbitrarily close Hausdorff approximation from inside by continua of the indicated class (Kihara, 2011).

In algebraic interpolation, the native term is unattainable point. For the Rational Hermite Interpolation Problem, these are exactly the interpolation data for which no solution exists. If

$M$06

then the set of unattainable points decomposes as

$M$07

Each $M$08 is equidimensional of codimension $M$09, and over an algebraically closed field each is a union of $M$10 rational irreducible varieties. The odd codimension reflects a two-step structure: first the weak interpolation kernel jumps in dimension, then the denominator vanishes at one interpolation node (Teresa et al., 2017).

7. Arithmetic and algorithmic variants

Arithmetic geometry supplies an instructive counterpoint: some apparently non-approachable points are only finitely distant from approachability. For a regular geometrically integral curve $M$11 in characteristic $M$12, a non-smooth non-decomposed point $M$13 becomes $M$14-rational after finitely many normalized Frobenius pullbacks. Writing

$M$15

the paper determines the exact worst-case number $M$16 of steps. Its main theorem shows that

$M$17

with a complete criterion deciding which value occurs. Thus the obstruction is not permanence but the sharp combinatorics of Frobenius descent (Hilario, 2024).

Diophantine approximation provides a different notion of resistance. A point $M$18 is badly approximable if there exists $M$19 such that

$M$20

for every $M$21 and $M$22. These are the points most difficult to approximate by rational vectors. For a planar non-linear carpet $M$23 satisfying the coordinate open set condition and having at least two maps in some column and some row, the paper proves

$M$24

So in this setting the most Diophantinely non-approachable points are not sparse in Hausdorff dimension (Anttila et al., 12 Mar 2026).

Optimization theory converts non-approachability into computational hardness. For cubic polynomials in $M$25 variables, if there were a polynomial-time algorithm that outputs a point with

$M$26

whenever the polynomial has a critical point, then $M$27. The paper also proves hardness for approximating critical points in Euclidean distance, for lower bounded functions, and in settings where existence or uniqueness of a critical point is guaranteed and where there are no spurious critical points. Here non-approachability is algorithm-independent: the obstruction is not a particular descent dynamics but worst-case intractability in the standard Turing model (Ahmadi et al., 29 Jan 2026).

Taken together, these literatures show that non-approachable points are not a single invariant class but a recurrent type of obstruction. Depending on the category, the obstruction may be metric, combinatorial, continuum-theoretic, geodesic, dynamical, constructive, arithmetic, or algorithmic. What remains stable is the underlying pattern: a point, subset, or datum is called non-approachable when the ambient theory supplies a canonical notion of access and the object systematically resists it.