Alphabeta MathThe Topology of Euclidean Space — Examples
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 3 results · all verified · 3 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 3 also cleared it.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
The Euclidean closed ball and sphere worked through the compactness equivalence chart
Counterexample
The open unit ball in ℝⁿ is bounded and not compact
Counterexample
ℝⁿ is closed and unbounded and is not compact for n≥1
Example
The straight segment between two points of an open Euclidean ball stays in the ball
Example
GL₁(ℝ)=ℝ∖{0} is disconnected, whereas ℝ²∖{0} is polygonally connected
Counterexample