Alphabeta MathArticle items · 03 — Natural transformation and its components
Presents 27 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Covariant functor, identity functor, composite functor, and contravariant functor
- Subgroup
- Conjugation x↦ gxg⁻¹ is an automorphism
- Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints
- Continuity of a map of topological spaces at a point and globally
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Well-order and well-ordered set
- Injection, surjection, bijection
- The Kuratowski ordered pair (a,b) := {{a},{a,b}}
- Refinements, locally finite families, point-finite families, and star refinements
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- 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
- 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
- Open ball, closed ball and sphere in a metric space
- Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not
- Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous
- The initial topology of a family of maps into spaces and the final topology of a family of maps out of spaces, and the subspace topology as the model initial topology
- Basis and subbasis for a topology, and the topology generated by a family of sets
- A family is a basis for a unique topology iff it covers the set and every point of an intersection of two members lies in a member inside that intersection; finite intersections of any subbasis form a basis
- Choice function
- The Axiom of Choice
- Every natural-number-indexed list of nonempty sets has a choice function on its family of values
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Normal subgroup: invariance under conjugation
- The coset set G/H and the index [G:H] of a subgroup
- Left and right cosets gH and Hg of a subgroup
- 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)
- Linear subspace of a vector space
- Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings
- Deck transformations and the deck-transformation group of a covering
- Binary operation on a set; associativity, commutativity, and a subset closed under the operation
- Left identity, right identity, and two-sided identity for a binary operation
- A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity
- The homomorphism on fundamental groups induced by a pointed continuous map
- Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open
- Based loops and the fundamental group
- The Axiom Schema of Separation: for each formula φ, ∀ p̄ ∀ x ∃ y ∀ z (z ∈ y ↔ (z ∈ x ∧ φ(z,p̄)))
- The Axiom of Extensionality: ∀ x ∀ y (∀ z (z ∈ x ↔ z ∈ y) → x = y)
- Monoid homomorphism and group homomorphism
- The kernel and image of a group homomorphism
- The first-order language of set theory: ∈, =, formulas with parameters, and class abbreviations
- The Axiom of Union: ∀ x ∃ y ∀ z (z ∈ y ↔ ∃ s (s ∈ x ∧ z ∈ s))
- Simply connected topological spaces
- The Axiom of Pairing: ∀ x ∀ y ∃ z ∀ t (t ∈ z ↔ (t = x ∨ t = y))
- Field
- Group and abelian group
- Left inverse, right inverse, and invertible element of a monoid
- In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided
- Every path-connected space is connected, and every path component lies inside a component
- For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one 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
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
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
DefinitionLibrary prerequisite
Natural transformation and its components
DefinitionLibrary prerequisite
The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup
DefinitionLibrary prerequisite
Nullhomotopic maps and contractible spaces
DefinitionLibrary prerequisite
Opposite category Cᵒᵖ
DefinitionLibrary prerequisite
Order embedding and order isomorphism
DefinitionLibrary prerequisite
Paracompactness: every open cover has a locally finite open refinement, with no separation axiom built into the word
DefinitionLibrary prerequisite
Partial order and partially ordered set
DefinitionLibrary prerequisite
Paths, path-connected spaces and path components
DefinitionLibrary prerequisite
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
DefinitionLibrary prerequisite
The quotient group G/N and coset product (gN)(hN)=ghN
DefinitionLibrary prerequisite
The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection
DefinitionLibrary prerequisite
The quotient vector space V/W and its canonical projection
DefinitionLibrary prerequisite
Regular coverings
DefinitionLibrary prerequisite
Semigroup and monoid
DefinitionLibrary prerequisite
Semilocally simply connected spaces with explicit basepoint convention
DefinitionLibrary prerequisite
The difference a ∖ b, the symmetric difference a △ b, and the complement X ∖ a relative to a set X
DefinitionLibrary prerequisite
Group extensions, sections, complements, and split extensions
DefinitionLibrary prerequisite
Subcategory and full subcategory
DefinitionLibrary prerequisite
Subset x ⊆ y, proper subset x ⊊ y, and the separation notation { z ∈ x : φ(z) }
DefinitionLibrary prerequisite
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
DefinitionLibrary prerequisite
Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
DefinitionLibrary prerequisite
The union ⋃ x of a set, and the binary union a ∪ b := ⋃ {a,b}
DefinitionLibrary prerequisite
Universal covering spaces
DefinitionLibrary prerequisite
The unordered pair {x,y} and the singleton {x} = {x,x}
DefinitionLibrary prerequisite
Vector space over a field
DefinitionLibrary prerequisite
Identity natural transformation and vertical composition
ExampleLibrary prerequisite