The Topology of Euclidean Space

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1 · Prerequisites

Abstract Algebra

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Linear Algebra

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Remark

For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent

Corollary

Assuming AC_ω and DC, compactness, sequential compactness, countable compactness, limit point compactness, completeness and total boundedness, pseudocompactness, closedness and boundedness, and the extreme-value property are equivalent for nonempty subsets of ℝⁿ with n≥1

Corollary

For n≥1, the map H(x,t)=((1-t)+t/‖ x‖₂)x is continuous on (ℝⁿ∖{0})×[0,1], starts at x, ends at radial normalisation, fixes the unit sphere, and never reaches 0

Corollary

The Topology of Euclidean Space

3 · Items

The product, Euclidean-metric and norm topologies on ℝⁿ agree, and for n=1 they agree with the real-line topology

Euclidean spheres and closed balls as subspaces of ℝⁿ

Pseudocompact space: every continuous real-valued function has bounded image

A pseudocompact subset of ℝⁿ is bounded

A pseudocompact subset of ℝⁿ is closed

For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent

Assuming AC_ω and DC, compactness, sequential compactness, countable compactness, limit point compactness, completeness and total boundedness, pseudocompactness, closedness and boundedness, and the extreme-value property are equivalent for nonempty subsets of ℝⁿ with n≥1

For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact

Polygonal paths and polygonally connected subsets of ℝⁿ

A finite concatenation of straight segments in ℝⁿ is a continuous path

The points polygonally reachable from a fixed point form a clopen subset of every open subset of ℝⁿ

For an open subset of ℝⁿ, connectedness, path-connectedness and polygonal connectedness are equivalent

Every connected component of an open subset of ℝⁿ is open and polygonally connected

ℝⁿ is polygonally connected, connected, locally path-connected and locally connected

ℝⁿ is locally compact and σ-compact

Every finite power of an at most countable set is at most countable

ℚⁿ is a countable dense subset of ℝⁿ, and rational open boxes form a countable basis

For n≥2, the punctured space ℝⁿ∖{0} is polygonally connected

Radial normalisation x↦ x/‖ x‖₂ is continuous on ℝⁿ∖{0}

For n≥1, the map H(x,t)=((1-t)+t/‖ x‖₂)x is continuous on (ℝⁿ∖{0})×[0,1], starts at x, ends at radial normalisation, fixes the unit sphere, and never reaches 0

For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected

ℝ is not homeomorphic to ℝⁿ for any n≥2

FALSE: every connected subset of ℝⁿ is polygonally connected