Alphabeta Math

Topology of R

About this page
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 14 results · all verified · 13 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ

Definition

Open subset of ℝ (every point has a neighbourhood inside it), closed subset (complement open), and clopen

Theorem

Arbitrary unions and finite intersections of open subsets of ℝ are open, and dually for closed sets

Definition

Interior, closure, boundary and exterior of a subset of ℝ

Definition

Limit point, isolated point, adherent point, derived set, and dense subset of ℝ

Theorem

The closure equals the set together with its limit points, equals the set of points every neighbourhood of which meets it, and is the smallest closed superset; a set is closed iff it contains its limit points

Lemma

A point lies in the closure of A ⊆ ℝ iff some sequence in A converges to it, so a subset of ℝ is closed iff it is sequentially closed

Theorem

Every open subset of ℝ is a countable disjoint union of open intervals, namely its order components

Lemma

Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable

Definition

Open cover, subcover, compact subset of ℝ (every open cover has a finite subcover), and sequentially compact subset

Theorem

Heine-Borel by bisection: every closed bounded interval [a,b] is compact

Lemma

A compact subset of ℝ is closed and bounded

Theorem

A subset of ℝ is compact if and only if it is closed and bounded

Theorem

A subset of ℝ is compact iff it is sequentially compact

Definition

Separated sets, disconnection, and connected subset of ℝ

Theorem

A subset of ℝ is connected if and only if it is order-convex, that is, an interval

Definition

Perfect subset of ℝ: closed with no isolated points

Theorem

Every nonempty perfect subset of ℝ is uncountable

Remark

Which results on this page use the order of ℝ and therefore have no general-topological analogue

False statement

FALSE: an arbitrary intersection of open subsets of ℝ is open

False statement

FALSE: every subset of ℝ is either open or closed

False statement

FALSE: in every ordered field a closed bounded set is compact, so Heine-Borel needs no completeness

Sources