Alphabeta Math

Topology of R: Examples and Counterexamples

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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.

✓ 11 results · all verified · 11 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; all 11 also cleared it.
1 · Prerequisites
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

An explicit open subset of ℝ written as the disjoint union of its component intervals

Example

ℚ has closure ℝ, empty interior, and boundary ℝ

Example

{1/k : k ≥ 1} ∪ {0} is compact while {1/k : k ≥ 1} is not closed

Example

Every nondegenerate closed interval is perfect, giving a second proof that it is uncountable

Counterexample

⋂ₖ (-1/k, 1/k) = {0} is not open

Counterexample

[0,1) is neither open nor closed in ℝ

Counterexample

The cover {(1/k, 1)} of (0,1) has no finite subcover, so (0,1) is not compact

Counterexample

{q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact

Counterexample

ℚ ∩ [0,2] is bounded and disconnected, so being an interval of ℚ is not enough

Counterexample

{0} ∪ [1,2] is closed, has an isolated point, and is not perfect

Counterexample

ℤ is closed and not compact, and (0,1) is bounded and not compact: neither hypothesis of Heine-Borel can be dropped

Sources