Alphabeta MathArticle items · 04 — In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Every natural-number-indexed list of nonempty sets has a choice function on its family of values
- Open cover, subcover, compact metric space, and compact subset of a metric space
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Isometry, isometric embedding, and the subspace metric on a subset
- Every subspace of a metrizable space is metrizable and every subspace of a first countable space is first countable, the metric case being the subspace metric already identified with the subspace topology
- Continuity of a map of topological spaces at a point and globally
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
- A continuous bijection is a homeomorphism iff it is open iff it is closed, and homeomorphy is an equivalence relation on spaces
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- Complete ordered field (least-upper-bound property)
- Order on the reals
- Maximum and minimum of a set
- Every nonempty finite set of reals has a maximum and a minimum
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- The reals form a totally ordered field
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Metric continuity characterisations, with countable choice for the sequential converse
- Injection, surjection, bijection
- Riemann's criterion: a bounded f on [a,b] is Darboux integrable if and only if for every real ε > 0 there is a partition P with U(f,P) - L(f,P) < ε
- Heine-Cantor in ℝ: a continuous real function on a compact subset of ℝ is uniformly continuous
- Uniform continuity of f : A → ℝ: one δ serving every pair of points of A
- A subset of ℝ is compact if and only if it is closed and bounded
- Open cover, subcover, compact subset of ℝ (every open cover has a finite subcover), and sequentially compact subset
- Open subset of ℝ (every point has a neighbourhood inside it), closed subset (complement open), and clopen
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- A continuous real function on a compact subset of ℝ is bounded
- Partition of [a,b] as a finite strictly increasing list a = t₀ < t₁ < … < tₙ = b, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions
- For bounded f on [a,b] and a partition P: the infimum mᵢ and supremum Mᵢ of f on the i-th subinterval, and the lower and upper Darboux sums L(f,P) = ∑ᵢ mᵢ Δᵢ and U(f,P) = ∑ᵢ Mᵢ Δᵢ
- The lower and upper Darboux integrals of a bounded f on [a,b] as sup_P L(f,P) and inf_P U(f,P), Darboux integrability as their equality, and the notation ∫ₐᵇ f
- Laws of finite sums and finite products
- Finite sums and finite products, by recursion
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- Every complete ordered field is Archimedean
- The canonical natural ι(n) = n · 1_F of a field
- Canonical naturals are positive and strictly increasing
- Lower bound, bounded below, bounded set
- Basic properties of the absolute value
- Ordered field
- Order is preserved by adding a constant and by adding inequalities
- Sign rules for products and monotonicity of multiplication
- Smoothly homotopic maps induce the same de rham map
- De rham cohomology is a contravariant functor
- Pullback is a homomorphism of de rham cohomology algebras
- Homotopy equivalences, homotopy inverses and spaces of the same homotopy type
- Mayer vietoris sequence in de rham cohomology
- Poincare lemma for differential forms on star shaped domains
- Zero th de rham cohomology is locally constant functions
- De rham cohomology of a finite disjoint union is the direct sum
- The real Frechet derivative on real and complex matrix spaces with the Frobenius norm
- The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix
- The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring
- Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring
- Laplace expansion computes the determinant along every row and every column over a commutative ring
- For every positive-sized square matrix over a commutative ring, Aadj(A)=adj(A)A=det(A)I
- The trace tr(A) as the sum of the diagonal entries
- Invertible matrices and the general linear group GLₙ(F)
- Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose
- Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products
- For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Reverse Fatou's lemma under an integrable majorant
- Integrable real and complex functions, and their integrals
- The Lebesgue integral is linear on L¹(μ)
- The modulus of an integral is bounded by the integral of the modulus
- Additivity of the nonnegative Lebesgue integral
- A nonnegative integral over a null set vanishes
- A nonnegative measurable function with finite integral is finite almost everywhere
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Local defining maps for embedded submanifolds
- Embedded submanifolds and slice charts
- Chart maps are diffeomorphisms onto Euclidean open sets
- The chain rule for differentials of smooth maps
- A pseudocompact subset of ℝⁿ is bounded
- A pseudocompact subset of ℝⁿ is closed
- Pseudocompact space: every continuous real-valued function has bounded image
- Complete metric space: every Cauchy sequence converges in the space
- The Cauchy criterion from the least-upper-bound property: in a complete ordered field every Cauchy sequence converges
- Cauchy sequence in a metric space
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- Limits and Cauchy sequences of reals
- Squaring is monotone on the nonnegatives
- Inverses of positives are positive, and reciprocation reverses order
- The well-ordering principle
- A sequence has at most one limit
- The rationals embed densely in the reals
- Smooth bundle metrics
- Local and global frames of a vector bundle
- Local frames and local trivializations are equivalent data
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- Smooth partitions of unity exist on manifolds
- The Axiom of Countable Choice (AC_ω)
- Every nonempty set bounded below has an infimum
- A compact subset of a metric space is closed and bounded
- Greatest lower bound (infimum)
- Epsilon characterisation of the supremum
- Epsilon characterisation of the infimum
- The closure of a nonempty A is {x : d(x,A) = 0}, equals A together with its limit points, and is the smallest closed superset
- Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
- Open ball, closed ball and sphere in a metric space
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- Tube lemma: if K is compact and an open N ⊆ X × Z contains K × {z₀}, then N contains K × W for some open W ∋ z₀
- The product set ∏_i ∈ I Xᵢ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- The principle of mathematical induction
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- The natural numbers ℕ (von Neumann)
- Fredholm operator cokernel and index
- Compact linear operator
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Banach space
- Atkinson
- Compositions with a compact operator are compact
- Linear combinations of compact operators are compact
- Fredholm index is locally constant
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- 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
- Separated sets, disconnection, and connected subset of ℝ
- The Axiom of Choice
- AC supplies the countable and dependent choices used in Banach integration
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- The indefinite integral of an L¹ function is absolutely continuous
- The indefinite integral of an L¹ function is differentiable almost everywhere
- Absolutely continuous functions have integrable derivatives
- An absolutely continuous function with zero derivative almost everywhere is constant
- Local and global flows generated by a vector field
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Through each point there is a unique maximal integral curve
- Compact-support Stokes on the upper half-space
- Compact-support Stokes on Euclidean space
- Localization of Stokes by a partition of unity
- Change of variables on oriented manifolds
- Integration on an oriented embedded submanifold
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
- Gauss-Bonnet for compact oriented surface regions with boundary and corners
- Regular oriented surface regions with corners
- Topological well-definedness of the surface Euler characteristic
- A nested sequence of nonempty closed bounded intervals has nonempty intersection, and the intersection is a single point exactly when the lengths tend to 0
- The recursion theorem
- The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ
- The multiplicative identity is positive
- A closed subset of a compact metric space is compact
- Integer powers aᵐ
- For |r| < 1 the sequence rᵏ is null, and for |r| > 1 the sequence |r|ᵏ diverges to +∞
- Absolute value in an ordered field
- Inverse function theorem for Banach spaces
- Chain sum product and composition rules for Banach derivatives
- The standard product norms on a finite product of normed spaces
- C k map between Banach spaces
- Fréchet derivative between Banach spaces
- Closed subspaces of complete metric spaces are complete; the converse under countable choice
- The spaces B(X,Y) and B(X) of bounded linear operators
- Neumann series and small perturbations of bounded inverses
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
- f is continuous at c ∈ A if and only if f(xₖ) → f(c) for every sequence in A converging to c, the converse direction costing countable choice
- Algebra of limits: sums, scalar multiples, products and quotients
- Limits preserve non-strict inequalities
- Sequences of reals: bounded, eventually, frequently, tails, subsequences
- Laws of integer exponents
- Smooth isotopies, diffeotopies and ambient isotopies
- The velocity field of an isotopy is well defined along its image
- The velocity field of an isotopy extends to a neighbourhood
- Compactness gives a compactly supported time-dependent velocity field
- A compactly supported time-dependent field has a global time-one flow
- Smooth embeddings
- Diffeomorphisms and local diffeomorphisms of manifolds
- A smooth Urysohn lemma for a closed set in an open set
- In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular
- Smooth maps between manifolds with boundary
- Interior and boundary of a manifold with boundary
- Smooth partitions of unity exist on manifolds with boundary
- Smooth maps are continuous
- |d(x,A) - d(y,A)| ≤ d(x,y), so the distance to a fixed nonempty set is 1-Lipschitz
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
TheoremLibrary prerequisite
In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
TheoremLibrary prerequisite
For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide
TheoremLibrary prerequisite
A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
TheoremLibrary prerequisite
For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant
TheoremLibrary prerequisite
The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
TheoremLibrary prerequisite
A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
TheoremLibrary prerequisite
De rham cohomology is smooth homotopy invariant
TheoremLibrary prerequisite
De rham cohomology of spheres
TheoremLibrary prerequisite
The determinant differential is D det(A)[H]=tr(adj(A)H) at every matrix, and Jacobi's formula holds on the invertible locus
TheoremLibrary prerequisite
Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
TheoremLibrary prerequisite
Dominated convergence
TheoremLibrary prerequisite
Embedded submanifolds admit local defining submersions
TheoremLibrary prerequisite
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
TheoremLibrary prerequisite
ℝ and ℝⁿ for n ≥ 1 with the Euclidean metric are complete, componentwise from the Cauchy criterion in ℝ
TheoremLibrary prerequisite
Every smooth vector bundle admits a smooth bundle metric
TheoremLibrary prerequisite
A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
TheoremLibrary prerequisite
A product of finitely many compact spaces is compact in the product topology
TheoremLibrary prerequisite
Fredholm index is stable under compact perturbations
TheoremLibrary prerequisite
Fundamental theorem of calculus for absolutely continuous functions
TheoremLibrary prerequisite
The fundamental theorem on flows
TheoremLibrary prerequisite
The general Stokes theorem
TheoremLibrary prerequisite
Global Gauss-Bonnet for closed oriented surfaces
TheoremLibrary prerequisite
Heine-Borel by bisection: every closed bounded interval [a,b] is compact
TheoremLibrary prerequisite
Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
TheoremLibrary prerequisite
Implicit function theorem for Banach spaces
TheoremLibrary prerequisite
Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
TheoremLibrary prerequisite
The isotopy extension theorem
TheoremLibrary prerequisite