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Article items · 05 — A point lies in the closure of $A$ iff every basic neighbourhood of it meets $A$; the closure is the smallest closed superset and equals $A$ together with its derived set
Presents 29 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
- Basis and subbasis for a topology, and the topology generated by a family of sets
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- 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
- 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
- Complex Lp classes and Euclidean test-function conventions
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Minkowski's inequality for integrals, including p = ∞
- The modulus of an integral is bounded by the integral of the modulus
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The essential supremum of a measurable function with respect to a measure
- Finite and countable subadditivity of measures
- Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞
- Assuming Countable Choice, Lᵖ-convergent sequences have almost-everywhere convergent subsequences
- The Axiom of Countable Choice (AC_ω)
- Open cover, subcover, compact metric space, and compact subset of a metric 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
- 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
- Isometry, isometric embedding, and the subspace metric on a subset
- Injection, surjection, bijection
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Reverse Fatou's lemma under an integrable majorant
- Integrable real and complex functions, and their integrals
- 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
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- The completed product measure
- Sobolev extension domains and extension operators
- Bounded Cᵏ domains and boundary charts
- Integer-order Sobolev extension from a half-space
- Cᵏ boundary flattening preserves local W^k,p
- Test function cutoffs and euclidean localization
- Weak Leibniz rule with a smooth factor
- Bounded restriction and cutoff localisation in Sobolev spaces
- Compactly supported Sobolev functions extend by zero in every integer order
- Integer-order Sobolev spaces and their norms
- Monotone convergence for the integral
- The Axiom of Choice
- Closure properties of measurable functions used by the integral
- Weighted tempered-distribution characterization of Hˢ
- Real-order Bessel-potential completion Hˢ
- The Bessel completion embeds canonically in tempered distributions
- Real powers of the Japanese bracket act on Schwartz space
- Smooth polynomially bounded multipliers on schwartz space
- Regular distribution from a locally integrable function
- Weak derivative of a locally integrable function
- Weak differentiation ignores null-set changes
- Uniqueness of a weak derivative as an almost-everywhere class
- The Sobolev norm descends to equivalence classes
- Distributional derivatives are polynomial Fourier multipliers
- Plancherel theorem
- Fourier differentiation and multiplication identities on tempered distributions
- Fourier transform agrees with l one and plancherel transforms
- Fourier transform is a topological automorphism of tempered distributions
- Polynomial growth functions define tempered distributions
- Locally integrable functions embed in distributions
- Euclidean Gaussian transform with the 2π normalization
- The exponential dominates every fixed nonnegative integer power at +∞
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- The class L¹(μ) of integrable functions
- A uniform limit of continuous functions is continuous, so C(X,Y) is closed in Y^X under the uniform metric
- Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)
- For a nonempty set X and a metric space (Y,d) the uniform metric ρ̄(f,g) = supₓ min{d(f(x),g(x)), 1} is a metric on Y^X
- min(d,1) and d/(1+d) are metrics uniformly equivalent to d, so every metric space carries a bounded metric with the same topology
- Complete metric space: every Cauchy sequence converges in the space
- Cauchy sequence in a metric space
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- Closed subspaces of complete metric spaces are complete; the converse under countable choice
- A sequence in a metric space has at most one limit
- Maximum and minimum of a set
- Every nonempty finite set of reals has a maximum and a minimum
- The topology of pointwise convergence on Y^X, which is the product topology, and its restriction to C(X,Y)
- The rationals embed densely in the reals
- Open ball, closed ball and sphere in a metric space
- Complete ordered field (least-upper-bound property)
- Suprema and infima are unique
- The supremum metric d_∞(f,g) = supₓ |f(x) - g(x)| is a metric on the bounded real-valued functions on a nonempty set
- A nonconstant complex polynomial tends to infinite modulus and attains a global minimum modulus
- A nonzero value of a nonconstant complex polynomial cannot be a local minimum of its modulus
- The koszul formula defines an affine connection
- Koszul formula is necessary for a levi civita connection
- Levi civita connection
- The admissible values in a one-step Hahn-Banach extension form a nonempty interval
- The union of a chain of dominated extensions is a well-defined dominated linear functional
- Zorn's lemma
- A compact subset of a metric space is closed and bounded
- A closed subset of a compact metric space is compact
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- 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
- 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
- Finite sums and finite products, by recursion
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- Integer powers aᵐ
- For |r| < 1 the sequence rᵏ is null, and for |r| > 1 the sequence |r|ᵏ diverges to +∞
- Laws of finite sums and finite products
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- 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
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- Squaring is monotone on the nonnegatives
- Absolute value in an ordered field
- The canonical natural ι(n) = n · 1_F of a field
- Limits and Cauchy sequences of reals
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- Uniform continuity of a map of metric spaces: one δ serving every point
- Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Conjugate exponents, including the endpoint conventions
- The function space Lᵖ(μ) for 0 < p < ∞
- The space L^∞(μ) of essentially bounded measurable functions
- The essential supremum is attained as the least essential bound
- Young's inequality for conjugate real exponents
- Measure-preserving transformations and systems
- Every nonnegative measurable function is the increasing limit of simple measurable functions
- The Lᵖ trace operator on a bounded C¹ domain
- The trace agrees with classical restriction for continuous Sobolev functions
- The trace commutes with smooth cutoffs and is chart local
- The Gauss-Green integration-by-parts formula with Sobolev traces
- Zero-boundary Sobolev space as a norm closure
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Compactly supported smooth functions are dense in W^k,p(Rⁿ)
- Finite ambient partitions near compact sets
- ‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Fundamental theorem of calculus for absolutely continuous functions
- Complex translation, convolution, approximate identities, and mollification
- The L1 transform is bounded and uniformly continuous
- Simultaneous L1 and L2 smooth approximation
- Complex completeness, density, and inner product: the consumer interface
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume
- Outer measures
- Axis-parallel rectangles in ℝᵐ and their volume
- Half-open boxes in ℝⁿ and their volume
- Measures on sigma-algebras
- Lebesgue outer measure on ℝⁿ
- Countable covers by closed boxes, by open boxes and by closed cubes all compute Lebesgue outer measure
- Laws of integer exponents
- Carathéodory measurable sets
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined
- Series in the nonnegative extended real line
- Basic properties of the absolute value
- Inverses of positives are positive, and reciprocation reverses order
- Sign rules for products and monotonicity of multiplication
- Ordered field
- Elementary sets: the finite unions of half-open boxes in ℝⁿ
- The sum of the volumes of a disjoint box decomposition of an elementary set does not depend on the decomposition
- Every elementary set is a finite disjoint union of half-open boxes, and any finitely many boxes admit a common grid refinement
- Translation of a subset of ℝⁿ
- Distributional derivative
- Multiplication of a distribution by a smooth function
- Kernel and image of a linear map
- Linear map between vector spaces over the same field
- One-step subspace test: a nonempty W ⊆ V is a linear subspace if and only if λ u + v ∈ W for all λ ∈ F and u, v ∈ W
- A linear map preserves zero, negatives, and subtraction
- Distribution
- Test function lf topology universal property
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
A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
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
Complex Holder, Minkowski, and the quotient norm
TheoremLibrary prerequisite
Complex Lp completeness and almost-everywhere subsequences
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
Dominated convergence
TheoremLibrary prerequisite
The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures
TheoremLibrary prerequisite
Bounded Cᵏ domains admit integer-order Sobolev extension
TheoremLibrary prerequisite
Fatou's lemma
TheoremLibrary prerequisite
Real-order Hˢ as weighted Fourier distributions
TheoremLibrary prerequisite
Integer-order W^k,2 and Hᵏ agree with equivalent norms
TheoremLibrary prerequisite
Fubini's theorem for L¹ functions on a sigma-finite product
TheoremLibrary prerequisite
If (Y,d) is complete then Y^X is complete in the uniform metric, and so is C(X,Y)
TheoremLibrary prerequisite
Fundamental theorem of algebra: every nonconstant complex polynomial has a complex root
TheoremLibrary prerequisite
Fundamental theorem of riemannian geometry
TheoremLibrary prerequisite
Hahn-Banach dominated extension theorem for real vector spaces
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
Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
TheoremLibrary prerequisite
Holder's inequality for integrals, including the endpoint cases
TheoremLibrary prerequisite
Integral invariance under measure-preserving maps
TheoremLibrary prerequisite
The kernel of the trace is the closure of the test functions
TheoremLibrary prerequisite
Agreement of the integral and L2 transforms
TheoremLibrary prerequisite
A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
TheoremLibrary prerequisite
For a nonzero real c, dilation by c multiplies Lebesgue outer measure by |c|ⁿ, and reflection in the origin preserves it
TheoremLibrary prerequisite
Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
TheoremLibrary prerequisite
Leibniz rule for distributions
TheoremLibrary prerequisite
The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial
TheoremLibrary prerequisite
The Lebesgue integral is linear on L¹(μ)
TheoremLibrary prerequisite