K-Topology¶

The topology on the real line generated by ordinary open intervals together with intervals having K={1/n} deleted, a canonical finer-than-Euclidean counterexample that is Hausdorff but not regular.

Version: v1 · 2026-08-30 · History
Domain-specific #: 2117
Origin domain: mathematics
Subdomain: general topology
Aliases: Smirnov Deleted Sequence Topology

Core Idea¶

The K-topology is a deliberately modified topology on the set of real numbers. Let

\[ K=\{1/n:n\in\mathbb N,\ n\ge 1\}. \]

Take as a basis every ordinary open interval \((a,b)\) together with every deleted-sequence interval \((a,b)\setminus K\). The topology generated by this basis is denoted \(\mathbb R_K\). Equivalently, start with the usual topology on \(\mathbb R\) and declare \(\mathbb R\setminus K\) open; intersections with usual intervals then produce the new basic neighborhoods.^[1]^[2]

This small intervention separates properties that familiar metric spaces encourage students to conflate. The new topology contains every Euclidean-open set, so it is strictly finer than the usual topology and remains Hausdorff. It also makes \(K\) closed, because its complement is now open. Yet the point \(0\notin K\) and the closed set \(K\) cannot be separated by disjoint open neighborhoods. Thus \(\mathbb R_K\) is Hausdorff but not regular, proving that the implication from regular to Hausdorff cannot be reversed.^[1]^[2]

The node is more than the sentence “there exists a counterexample.” It preserves the construction and its proof engine: a sequence accumulates at a point in the old topology; the new basis deletes that sequence from neighborhoods of the accumulation point, making the sequence closed; neighborhoods of individual sequence points remain ordinary and must reach back into every deleted neighborhood near the limit. That coupling makes point-to-point separation survive while point-to-closed-set separation fails.

Structural Signature¶

The signature is:

usual real line + deleted sequence \(K=\{1/n\}\) accumulating at \(0\notin K\) + basis \(\{(a,b),(a,b)\setminus K\}\) → a strict refinement in which \(K\) is closed but \(0\) and \(K\) have no disjoint neighborhoods

The mandatory roles are:

Carrier: the underlying set is \(\mathbb R\), not an arbitrary homeomorphic copy unless the construction is transported explicitly.
Distinguished sequence: \(K=\{1/n:n\ge1\}\), which converges to \(0\) in the usual topology but excludes its usual limit.
Baseline topology: ordinary Euclidean open intervals remain basic open sets.
Deletion basis: every \((a,b)\setminus K\) is also basic. In particular, deleted intervals form a local basis at \(0\).
Generated topology: arbitrary unions of basis members determine the open sets; the basis, not merely the intent to “make K closed,” fixes the object.
Refinement relation: the identity map \(\mathbb R_K\to\mathbb R\) with the usual topology is continuous, whereas the reverse identity is not.
Closed witness: \(K\) is closed in \(\mathbb R_K\) because \(\mathbb R\setminus K\) is open.
Separation witness: \(0\) and \(K\) cannot receive disjoint open neighborhoods, despite distinct points being Hausdorff-separable.

At points \(x\ne0\), the local topology agrees with the usual one. At a point of \(K\), a basis member containing that point must be an ordinary interval, because deleted intervals omit all of \(K\). At a point outside \(K\cup\{0\}\), a sufficiently small ordinary interval avoids \(K\). The modification is therefore concentrated in the neighborhood filter at \(0\), even though its global consequences include nonregularity, nonmetrizability, changed compactness, and changed path behavior.

What It Is Not¶

The K-topology is not the usual topology on \(\mathbb R\). In the usual topology, \(1/n\to0\) and \(K\) is not closed. In \(\mathbb R_K\), the open neighborhood \(\mathbb R\setminus K\) contains \(0\) and no term of that sequence, so the sequence does not converge to \(0\).

It is not the lower-limit or Sorgenfrey topology generated by half-open intervals \([a,b)\). Both are strict refinements of the usual real line, but the lower-limit and K-topologies are not comparable: a deleted neighborhood of \(0\) contains no sufficiently small half-open neighborhood of \(0\), while \([0,b)\) contains no K-basic neighborhood of \(0\).^[2]

It is not the particular-point, excluded-point, discrete, or cofinite topology. Those use global membership rules. The K-topology preserves ordinary local structure away from one accumulation point and is keyed to a specific deleted sequence.

It is not merely “the topology obtained by making \(K\) closed” without further qualification. Many finer topologies make \(K\) closed, including the discrete topology. The K-topology is the least topology generated from the Euclidean one by adjoining \(\mathbb R\setminus K\), equivalently the topology generated by the stated two-family basis.

It is not a k-topology in the distinct compactly generated or Kelleyfication sense, where open or closed behavior is tested on compact subspaces. Case and context matter: “Smirnov's deleted sequence topology” is the safer unambiguous surface for this object.

Finally, it is not a general recipe whose every substituted sequence automatically preserves all advertised properties. The proof relies on the relationship between \(K\), its omitted accumulation point, and the retained Euclidean neighborhoods. Generalizations require their own hypotheses.

Scope of Application¶

The K-topology is used in general topology to test separation axioms, metrizability implications, compactness, quotient constructions, and the effect of refining a topology. It is a standard counterexample separating Hausdorff spaces from regular spaces: every metric space is regular, so the same witness also proves that \(\mathbb R_K\) is not metrizable.^[1]

It supports analysis of quotient failure. If the closed subset \(K\) is collapsed to a point, a Hausdorff quotient would provide disjoint neighborhoods of the collapsed point and the image of \(0\); their inverse images would separate \(K\) and \(0\) in \(\mathbb R_K\), which is impossible. Thus “closed equivalence class” plus Hausdorff source is not sufficient to guarantee a Hausdorff quotient; regularity-type hypotheses matter.

The space also appears outside elementary exposition. De Brecht uses it as a simple nonmetrizable quasi-Polish space: the usual real line is Polish, \(\mathbb R\setminus K\) is a suitable \(\Delta^0_2\) set, and adjoining it as open preserves quasi-Polishness while producing a Hausdorff but nonregular, hence non-Polish, topology.^[3] This use shows that the construction can act as a boundary object in descriptive set theory rather than only as a classroom curiosity.

The scope remains mathematical. “Deleting troublesome cases from a neighborhood” may suggest analogies elsewhere, but the exact mechanism depends on open-set bases, convergence, separation axioms, and quotient topology. Those abstractions can transfer; the named K-topology does not leave topology intact.

Clarity¶

The example forces four claims to be audited separately: finer, Hausdorff, regular, and metrizable. Adding open sets makes the topology finer, but refinement does not monotonically improve every desirable property. Hausdorffness survives here because every Euclidean separation remains available. Regularity fails because the newly closed set \(K\) cannot be separated from \(0\). Metrizability then fails because metric spaces are regular. The construction therefore blocks the invalid shortcut “more open sets means better separation in every sense.”

It also clarifies that closedness and convergence are relative to a topology. The same subset \(K\) is nonclosed in the usual real line and closed in \(\mathbb R_K\); the same numerical sequence converges to \(0\) in one and not the other. No point or sequence changed—only the neighborhood system did.

Manages Complexity¶

Rather than searching a large exotic space for a separation counterexample, the K-topology localizes the entire failure around one familiar convergent sequence. Most points keep ordinary neighborhoods. One new open complement changes precisely the needed closure and convergence facts, while the retained Euclidean intervals preserve Hausdorff separation. The proof burden compresses to tracking how neighborhoods of \(0\) and of \(1/n\) intersect.

The object is therefore a compact test bench. It allows a student or researcher to test a proposed theorem by asking what the theorem predicts for a second-countable, Hausdorff, nonregular refinement of the real line. When a proof silently uses a metric, regularity, or point–closed-set separation, \(\mathbb R_K\) often reveals the missing hypothesis.

Abstract Reasoning¶

Basis verification. The two kinds of sets cover \(\mathbb R\). If a point lies in the intersection of two basis members, an ordinary interval around it can be narrowed and, when needed, have \(K\) deleted, producing a basis member inside the intersection. Hence the family generates a topology.

Refinement inference. Every Euclidean interval remains basic, so every Euclidean-open set is K-open. The deleted neighborhood \((-1,1)\setminus K\) is not Euclidean-open at \(0\), proving strictness. Consequently the identity from the refined space to the usual line is continuous, but not conversely.

Convergence diagnostic. For \(x\ne0\), K-topological and Euclidean convergence to \(x\) agree because local neighborhood bases agree. A sequence \((x_m)\) converges to \(0\) in \(\mathbb R_K\) exactly when it converges to \(0\) ordinarily and is eventually outside \(K\); the open neighborhood \(\mathbb R\setminus K\) supplies the necessary eventual-avoidance test. Thus \(1/n\not\to0\) in \(\mathbb R_K\).

Nonregularity witness. Suppose open \(U\ni0\) and open \(V\supseteq K\) were disjoint. Choose \((a,b)\setminus K\subseteq U\) around \(0\). For sufficiently large \(n\), \(1/n\in(a,b)\). Because \(1/n\in V\cap K\), an ordinary interval \((c,d)\subseteq V\) contains \(1/n\). That interval necessarily contains real points near \(1/n\) that are not in \(K\) and still lie in \((a,b)\); such a point belongs to both \(U\) and \(V\), a contradiction.^[1]^[4]

Metrizability inference. Every metrizable space is regular. Since \(\mathbb R_K\) is not regular, no metric induces its topology. This is stronger than merely noting that the ordinary Euclidean metric no longer works.

Compactness diagnostic. The identity \(\mathbb R_K\to\mathbb R\) is a continuous surjection. If \(\mathbb R_K\) were compact, its Euclidean image would be compact, contradicting noncompactness of the usual real line. More locally, any K-topological subspace containing all of \(K\) has an infinite closed discrete subset with no accumulation point and so cannot be compact under the standard compactness consequences for Hausdorff spaces.

Knowledge Transfer¶

Within topology, the complete mechanism transfers as a proof pattern: begin with a familiar space, select a nonclosed set missing an accumulation point, adjoin its complement as open, then audit which properties survive and which fail. This pattern helps construct countermodels for claims about refinement, convergence, separation, quotient spaces, and metrizability. The K-topology is the canonical concrete instance whose calculations can be replayed before attempting a broader construction.

Only the higher-level lessons transfer outside this exact object. Refinement can preserve one guarantee while breaking another; local rule changes can have global consequences; a counterexample should violate the target conclusion while preserving every stated hypothesis. Those belong to Refinement, Local-to-Global Reasoning, Counterexample, or Boundary. The mathematical object \(\mathbb R_K\), with its basis and separation proofs, remains domain-bound.

Examples¶

Formal/abstract¶

Take \(U=(-1/2,1/2)\setminus K\). It is a K-open neighborhood of \(0\). No term \(1/n\) lies in \(U\), although infinitely many terms lie in every ordinary interval around \(0\). This one set proves simultaneously that the new topology is strictly finer, that \(1/n\) no longer converges to \(0\), and that \(K\) is closed when the same deletion is applied globally as \(\mathbb R\setminus K\).

Now try to separate \(0\) from \(K\). Any K-neighborhood of \(0\) contains a deleted interval \((-\epsilon,\epsilon)\setminus K\). A neighborhood of a sufficiently small \(1/n\) must contain an ordinary interval around \(1/n\), since deleted intervals omit that point. No ordinary interval around \(1/n\) can avoid all nearby non-K points of the deleted interval. The intersection is forced, so regularity fails even though any two points can still use ordinary disjoint intervals.

Applied/practice¶

Suppose a proof claims: “If \(X\) is Hausdorff and \(A\subseteq X\) is closed, then collapsing \(A\) to one point yields a Hausdorff quotient.” Set \(X=\mathbb R_K\) and \(A=K\). Both hypotheses hold. If the quotient separated the collapsed point \([K]\) from \([0]\), the preimages of those disjoint quotient neighborhoods would be disjoint open neighborhoods of \(K\) and \(0\), contradicting the defining witness. The counterexample identifies the missing separation strength rather than merely disproving the statement.

In descriptive set theory, the same space tests the boundary between Polish and quasi-Polish spaces. Adjoining the \(\Delta^0_2\) set \(\mathbb R\setminus K\) as open produces a quasi-Polish topology, yet nonregularity prevents metrizability and therefore Polishness.^[3] The construction distinguishes the generalized completeness framework from the classical metrizable one with a familiar carrier and an explicit added open set.

Structural Tensions¶

Finer topology versus compactness. More open sets can make more distinctions available, but they also create more open covers and stricter convergence requirements. Refinement can preserve Hausdorffness while destroying compactness.

Point separation versus set separation. Ordinary intervals still separate any two points, yet the collective tail of \(K\) presses arbitrarily near \(0\), preventing disjoint neighborhoods of one point and one closed set. Hausdorff and regular separation are genuinely different quantifier structures.

Local concentration versus global consequence. Neighborhoods change materially only at \(0\), but the whole space becomes nonmetrizable and quotient behavior changes. A small local perturbation can alter global class membership.

Closure gained versus convergence lost. Declaring \(K\) closed removes its old limit relationship with \(0\). The design goal and side effect are the same topological change read in two vocabularies.

Familiar carrier versus unfamiliar space. The point set and numerical coordinates remain \(\mathbb R\), tempting Euclidean intuitions that no longer follow. The pair \((\mathbb R,\tau_K)\), not the carrier alone, is the object.

Structural–Framed Character¶

The K-topology is structural within mathematics: its definition uses only a carrier, a distinguished subset, a basis, unions, intersections, and neighborhood-separation relations. Its consequences are neutral formal entailments rather than institutional or evaluative judgments. Nevertheless, the vocabulary does not travel without topological interpretation. Open set, basis, refinement, convergence, Hausdorff, regular, quotient, and metrizable are mathematical commitments.

It is therefore a domain-specific abstraction, not a prime. Its counterexample logic is highly reusable inside topology and related formal work, while its substrate-neutral residue is already represented by more general catalog abstractions.

Structural Core vs. Domain Accent¶

The liftable core is a controlled counterexample: preserve a weaker property, alter one local rule, and force failure of a stronger property, thereby proving an implication strict. That reasoning can instantiate Counterexample, Refinement, Boundary, and Local-to-Global effects.

The irreducible domain accent is the exact topology \(\tau_K\): the real-line carrier, deleted reciprocal sequence, generated basis, neighborhood filters, and separation axioms. Strip those away and one can no longer derive that \(K\) is closed, \(1/n\) fails to converge to \(0\), Hausdorffness survives, or regularity fails. The remaining generic lesson is useful but already cataloged; the full proof engine warrants the domain node.

The K-topology instantiates Topology through an explicit open-set structure and Refinement by strictly enlarging the Euclidean topology. It uses Set and Membership, Union, Intersection, and Closure in the basis-generation machinery. It exhibits a local-to-global effect because a changed neighborhood filter at one accumulation point changes metrizability and quotient behavior for the whole space.

Its closest domain parent is Topological Space: \(\mathbb R_K\) is a particular pair \((X,\tau)\) satisfying the topology axioms. Open Set, Closed Set, Connectedness, and Compactness are properties or analytic lenses used on it, not exact coverage.

Relationships to Other Abstractions¶

Current abstraction K-Topology Domain-specific

Parents (1) — more general patterns this builds on

K-Topology is a kind of Topological Space Domain-specific

The K-topology instantiates Topology through an explicit open-set structure and Refinement by strictly enlarging the Euclidean topology.

Hierarchy paths (5) — routes to 3 parentless roots

K-Topology → Topological Space → Closure

Show alternative paths (4)

Neighborhood in Abstraction Space¶

K-Topology sits in a sparse region of the domain-specific corpus (77^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Usual real line: same carrier, coarser Euclidean topology; \(K\) is not closed and \(1/n\to0\).
Sorgenfrey or lower-limit line: generated by \([a,b)\), not by deleting \(K\); the two refinements are incomparable.
Discrete real line: every subset is open; far finer and regular.
Excluded-point topology: a global rule involving one point, not a localized deleted-sequence refinement.
A generic topology containing \(\mathbb R\setminus K\): may be finer still; the K-topology is the generated least extension with the stated basis.
Compactly generated k-topology/Kelleyfication: a different construction based on compact subspaces; not an alias despite overlapping typography.
K-theory: a family of algebraic-topological invariants, entirely unrelated to this K-named point-set topology.
A generalized deleted-set topology: substitutions for \(K\) require new hypotheses and may not preserve the same separation or connectedness results.

References¶

[1] Munkres, James R. Topology. 2^nd ed. Prentice Hall, 2000, p. 82 and §31, Example 1. registry ↩a ↩b ↩c ↩d

[2] Fjelstad, Jens. Notes on Topology, Examples 2.12–2.14 and Proposition 6.4. https://fuglede.dk/maths/teaching/basic-topology/FjelstadNotes.pdf registry ↩a ↩b ↩c

[3] de Brecht, Matthew. “Descriptive set theory of complete quasi-metric spaces.” RIMS Kôkyûroku 1790 (2012): 16–30, especially Theorem 43 and the K-topology example. https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1790-03.pdf registry ↩a ↩b

[4] Gardner, Robert. “Section 31: The Separation Axioms,” notes following Munkres. East Tennessee State University. https://faculty.etsu.edu/gardnerr/5357/notes/Munkres-31.pdf registry ↩

[5] Steen, Lynn Arthur, and J. Arthur Seebach Jr. Counterexamples in Topology. Dover reprint, 1995, Counterexample 64. registry

[6] Willard, Stephen. General Topology. Dover, 2004, Example 14.2. registry