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Set-theoretic function
A {\displaystyle A} is a subset of B , {\displaystyle B,} then the inclusion map is the function ι {\displaystyle \iota } that sends each element x {\displaystyle
Inclusion_map
Topics referred to by the same term
subset Inclusion (Boolean algebra), the Boolean analogue to the subset relation Inclusion map, or inclusion function, or canonical injection Inclusion (logic)
Inclusion
Subset of a manifold that is a manifold itself; an injective immersion into a manifold
S} which itself has the structure of a manifold, and for which the inclusion map S → M {\displaystyle S\rightarrow M} satisfies certain properties. There
Submanifold
Function that preserves distinctness
f:A\hookrightarrow B} ), although some authors specifically reserve ↪ for an inclusion map. For visual examples, readers are directed to the gallery section. For
Injective_function
Type of group in abstract algebra
exotic inclusion map S5 → S6 as a transitive subgroup (the obvious inclusion map Sn → Sn+1 fixes a point and thus is not transitive) and, while this map does
Symmetric_group
Tool in symplectic geometry
is also Hamiltonian, with momentum map the composition of the inclusion map with M {\displaystyle M} 's momentum map. Noether's theorem admits a particularly
Momentum_map
Counting technique in combinatorics
In combinatorics, the inclusion–exclusion principle (Commonly referred to as PIE) is a counting technique which generalizes the familiar method of obtaining
Inclusion–exclusion_principle
Induced map between the dual spaces of the two vector spaces
is an isometry if X {\displaystyle X} is a Banach space. Denote the inclusion map by In : M → X where In ( m ) := m for all m ∈ M . {\displaystyle
Transpose_of_a_linear_map
Most general completion of a commutative square given two morphisms with same domain
subspace of Y and g : Z → Y is the inclusion map we can "glue" Y to another space X along Z using an "attaching map" f : Z → X. The result is the adjunction
Pushout_(category_theory)
Finest topology making some functions continuous
function, namely the quotient map. The disjoint union topology is the final topology with respect to the inclusion maps. The final topology is also the
Final_topology
Mathematical description of spacetime used in relativity
\iota :\mathbf {H} _{R}^{1(n)}\rightarrow \mathbf {M} ^{n+1}} is the inclusion map and the superscript star denotes the pullback. The present purpose is
Minkowski_spacetime
Function that returns its argument unchanged
is idempotent. Every map from a set of a single element to itself is necessarily the identity map. Identity matrix Inclusion map Indicator function Knapp
Identity_function
Theorem in algebraic topology
are as above, we say that U {\displaystyle U} can be excised if the inclusion map of the pair ( X ∖ U , A ∖ U ) {\displaystyle (X\setminus U,A\setminus
Excision_theorem
Concept in homotopy theory
is said to be well-pointed if the inclusion map x → X {\displaystyle {x}\to X} is a cofibration. The inclusion map S n − 1 → D n {\displaystyle S^{n-1}\to
Cofibration
Function whose actual domain of definition may be smaller than its apparent domain
general convention, and the latter notation is more commonly used for inclusion maps or embeddings.[citation needed] Specifically, for a partial function
Partial_function
Scheme theory concept
interpreted by means of the fiber product of schemes, applied to f and the inclusion map of Y ′ {\displaystyle Y'} into Y. For the second, the idea is that morphisms
Flat_morphism
Inherited topology
Y → X {\displaystyle i:Y\to X} be the inclusion map. Then for any topological space Z {\displaystyle Z} a map f : Z → Y {\displaystyle f:Z\to Y} is continuous
Subspace_topology
Malaysia-headquartered policy network
The Alliance for Financial Inclusion (AFI) is a policy leadership alliance owned and led by member central banks and financial regulatory in developing
Alliance for Financial Inclusion
Alliance_for_Financial_Inclusion
"Smallest" commutative algebra that contains a vector space
linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g : S(V) → A such that f = g ∘ i, where i is the inclusion map of V
Symmetric_algebra
List of early depictions of the world
mappa mundi genre. The world map, as well as a map of the Holy Land and plan of Acre and Jerusalem were made for inclusion in Marino Sanuto's Liber Secretorum
Early_world_maps
Property in algebraic topology
{\displaystyle (X,A)} has the homotopy extension property, then the simple inclusion map ι : A → X {\displaystyle \iota \colon A\to X} is a cofibration. In fact
Homotopy_extension_property
Association of one output to each input
to f(X) that maps x to f(x). For every subset A of a set X, the inclusion map of A into X is the injective (see below) function that maps every element
Function_(mathematics)
Mathematical object in category theory
the inclusion map. Indeed, the subset Y := { x ∈ X ∣ χ ( x ) = 1 } {\displaystyle Y:=\{x\in X\mid \chi (x)=1\}} , equipped with the inclusion map Y →
Subobject_classifier
{O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}} is surjective. An example is the inclusion map Spec ( R / I ) → Spec ( R ) {\displaystyle \operatorname {Spec}
Closed_immersion
Topological vector spaces
denote the inclusion map by In K L : C k ( K ) → C k ( L ) . {\displaystyle \operatorname {In} _{K}^{L}:C^{k}(K)\to C^{k}(L).} Then this map is a linear
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Aspect of algebraic topology
\{U_{i}\}_{1\leq i\leq k}} of X {\displaystyle X} with the property that each inclusion map U i ↪ X {\displaystyle U_{i}\hookrightarrow X} is nullhomotopic. For
Lusternik–Schnirelmann category
Lusternik–Schnirelmann_category
On the finite presentability of fundamental groups of 3-manifolds
three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means
Scott_core_theorem
Surjective homomorphism
of monoids, Mon, the inclusion map N → Z is a non-surjective epimorphism. To see this, suppose that g1 and g2 are two distinct maps from Z to some monoid
Epimorphism
Generalization of the kernel of a homomorphism
→ Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical
Kernel_(category_theory)
complexes and simplicial maps, where the morphisms (i.e., relations in the poset) in the source category induce inclusion maps among the complexes. Note
Vietoris–Rips_filtration
following commutative diagram: Here i is the inclusion map and ΦX, ΦY are the maps obtained by composing the quotient map with the canonical injections into the
Adjunction_space
Coarsest topology making certain functions continuous
subspace with respect to the inclusion map. The product topology is the initial topology with respect to the family of projection maps. The inverse limit of
Initial_topology
If the inclusion i:H → G extends to a continuous map ∂i: ∂H → ∂G between their hyperbolic boundaries, the map ∂i is called a Cannon–Thurston map. Here
Cannon–Thurston_map
Ninth letter in the Greek alphabet
imaginary unit, but more often Roman i or j is used. In mathematics, the inclusion map of one space into another is sometimes denoted by the lowercase iota
Iota
Theorem of dominion in abstract algebra
epimorphisms. For example, let U be a subsemigroup of S containing U, the inclusion map U ↪ S {\displaystyle U\hookrightarrow S} is an epimorphism if and only
Isbell's_zigzag_theorem
well as in Southern Africa, to map the distribution of different diamond source regions. Sub-lithospheric mineral inclusions such as majorite and silicate
Diamond_inclusions
Inclusion of one mathematical structure in another, preserving properties of interest
f:X\hookrightarrow Y.} (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X {\displaystyle X} and Y {\displaystyle Y} , several different
Embedding
{\displaystyle T_{W}} is also self-adjoint. When V is replaced by the inclusion map I : W → H {\displaystyle I:W\to H} , V ∗ = I ∗ = P K : H → W {\displaystyle
Compression (functional analysis)
Compression_(functional_analysis)
a multivalued map, i.e. F(t, x) is a set rather than a single point in R d {\displaystyle \mathbb {R} ^{d}} . Differential inclusions arise in many situations
Differential_inclusion
Result of commutative algebra
affine space A k d {\displaystyle \mathbb {A} _{k}^{d}} . Then the inclusion map S ↪ A {\displaystyle S\hookrightarrow A} induces a surjective finite
Noether_normalization_lemma
Type of continuous linear operator
embedding when it is continuous. The embedding is called compact if this inclusion map is a compact operator; that is, if every bounded sequence in X {\displaystyle
Compact_operator
Way to extend a non-compact topological space
is used for the inclusion map c : X → X ∗ . {\displaystyle c:X\to X^{*}.} The properties below follow from the above discussion: The map c is continuous
Alexandroff_extension
Unsolved problem in geometry
dimension k, and let i : Z → X {\displaystyle i\colon Z\to X} be the inclusion map. Choose a differential form α {\displaystyle \alpha } of type ( p ,
Hodge_conjecture
Group that is also a differentiable manifold with group operations that are smooth
Lie group that is a subset of G {\displaystyle G} and such that the inclusion map from H {\displaystyle H} to G {\displaystyle G} is an injective immersion
Lie_group
can be homotoped into homotopies in the subset A. For example, for an inclusion map A ↪ X {\displaystyle A\hookrightarrow X} to be 1-connected, it must
Homotopical_connectivity
Notion in topology
the subspace topology. This is the coarsest topology which makes the inclusion map continuous. The natural topology on a quotient of a topological space
Natural_topology
Theorem about metric spaces
\rightarrow E} denote the identity (inclusion) map and let g : Ω → E {\displaystyle g:\Omega \rightarrow E} be a Lipschitz map of constant k < 1 {\displaystyle
Banach_fixed-point_theorem
Riemannian submanifold of R n + 1 {\displaystyle \mathbb {R} ^{n+1}} via the inclusion map S n ↪ R n + 1 {\displaystyle S^{n}\hookrightarrow \mathbb {R} ^{n+1}}
Riemannian_submanifold
Set whose elements all belong to another set
of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as
Subset
Concepts in commutative algebra
In the case of ordinary ring extensions such as A ⊆ B, the inclusion map is the pertinent map. The usual statements of going-up and going-down theorems
Going_up_and_going_down
Algebraic tool for computing topological spaces' invariants
l:B\hookrightarrow X} are inclusion maps and ⊕ {\displaystyle \oplus } denotes the direct sum of abelian groups. The boundary maps ∂ ∗ {\displaystyle \partial
Mayer–Vietoris_sequence
Normed vector space that is complete
canonically a metric Banach manifold modeled on X {\displaystyle X} since the inclusion map U → X {\displaystyle U\to X} is an open local homeomorphism. Using Hilbert
Banach_space
set of connected components of a space Y {\displaystyle Y} , and each inclusion map Z → Y {\displaystyle Z\to Y} induces a function π 0 ( Y ) → π 0 ( Z
End_(topology)
Function with a smaller domain
to a subset A {\displaystyle A} of X {\displaystyle X} is just the inclusion map from A {\displaystyle A} into X . {\displaystyle X.} The restriction
Restriction_(mathematics)
Group of symmetries of an n-dimensional hypercube
}} as a group of permutations of a set of size 2n induces a natural inclusion map ι : S n ± → S 2 n {\displaystyle \iota :S_{n}^{\pm }\to S_{2n}} from
Hyperoctahedral_group
Nonlinear differential operator used to study conformal mappings
group homomorphisms and Lie algebra homomorphisms leads to the "van Est inclusion map" H 1 ( Diff ( S 1 ) ; F λ ( S 1 ) ) ↪ H 1 ( Vect ( S 1 ) ; F λ (
Schwarzian_derivative
Right inverse of a morphism
a retraction in the topological sense, if it's a retraction of the inclusion map i : Y ↪ X {\displaystyle i:Y\hookrightarrow X} in the category theory
Section_(category_theory)
Establishes the concept of stabilization of homotopy groups
intersection is X {\displaystyle X} . Then, homotopy excision says the inclusion map: ( ( C X ) + , X ) ⊂ ( Σ X , ( C X ) − ) {\displaystyle ((CX)_{+},X)\subset
Freudenthal suspension theorem
Freudenthal_suspension_theorem
Construction for adding objects to a Hilbert space
which the inclusion map i : Φ → H , {\displaystyle i:\Phi \to H,} is continuous. Identifying H with its dual space H*, the adjoint to i is the map i ∗ : H
Rigged_Hilbert_space
1513 Ottoman nautical chart
The Piri Reis map is a world map compiled in 1513 by the Ottoman admiral and cartographer Piri Reis. Approximately one third of the map survives, housed
Piri_Reis_map
Where disabled students spend most of their time with non-disabled students
Inclusion in education refers to including all students to equal access to equal opportunities of education and learning, and is distinct from educational
Inclusion_(education)
Properties that homology theories of topological spaces have in common
such that the closure of U is contained in the interior of A, then the inclusion map i : ( X ∖ U , A ∖ U ) → ( X , A ) {\displaystyle i\colon (X\setminus
Eilenberg–Steenrod_axioms
Topological space with a distinguished point
X} which shares its basepoint with X {\displaystyle X} so that the inclusion map is basepoint preserving. One can form the quotient of a pointed space
Pointed_space
Type of morphism
normal subgroup of G. In particular, if H is a subgroup of G, then the inclusion map i from H to G is a monomorphism, and will be normal if and only if H
Normal_morphism
Notion of convergence in mathematics
identified as a subset of this Cartesian product via the canonical inclusion map F → ∏ x ∈ X Y {\displaystyle {\mathcal {F}}\to \prod _{x\in X}Y} defined
Pointwise_convergence
Concept in topology
N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps M ↪ W and N ↪ W {\displaystyle M\hookrightarrow W\quad {\mbox{and}}\quad
H-cobordism
Properties of mathematical functions
image, and let i : f ( X ) → Y {\displaystyle i\colon f(X)\to Y} be the inclusion map from f ( X ) {\displaystyle f(X)} into Y {\displaystyle Y} . Then f
Bijection, injection and surjection
Bijection,_injection_and_surjection
Generalization of the concept of subsequence to the case of nets
since the inclusion map ι : N → I {\displaystyle \iota :\mathbb {N} \to I} (that sends n ↦ n {\displaystyle n\mapsto n} ) is an order-preserving map whose
Subnet_(mathematics)
Principal bundle associated to a vector bundle
F ( E ) {\displaystyle F(E)} is the final topology coinduced by the inclusion maps π − 1 ( U i ) → F ( E ) {\displaystyle \pi ^{-1}(U_{i})\to F(E)} . With
Frame_bundle
Two continuous functions can be glued together to create another continuous function
, … . {\displaystyle X_{1},X_{2},X_{3},\ldots .} For instance, the inclusion map ι : Z → R {\displaystyle \iota :\mathbb {Z} \to \mathbb {R} } from the
Pasting_lemma
Structure preserving map derived canonically from another map
is a strong deformation retract of a topological space X, then the inclusion map from A to X induces an isomorphism between fundamental groups (so the
Induced_homomorphism
nuclear spaces. The inclusion map In D : X D → X {\displaystyle \operatorname {In} _{D}:X_{D}\to X} is called the canonical map. Suppose that D {\displaystyle
Auxiliary_normed_space
Map between simplicial sets with lifting property
{\displaystyle i} is the inclusion of Λ k n {\displaystyle \Lambda _{k}^{n}} in Δ n {\displaystyle \Delta ^{n}} ), there exists a map x : Δ n → X {\displaystyle
Kan_fibration
About direct sums and exact sequences
alternating subgroup, and let C = B/A ≅ {±1}. Let q and r denote the inclusion map and the sign map respectively, so that 0 ⟶ A ⟶ q B ⟶ r C ⟶ 0 {\displaystyle 0\longrightarrow
Splitting_lemma
Submodule of a mathematical ring
extension of A {\displaystyle A} , and we let f {\displaystyle f} be the inclusion map from A {\displaystyle A} to B {\displaystyle B} . The behaviour of a
Ideal_(ring_theory)
if U ⊆ M {\displaystyle U\subseteq M} is an open submanifold, with inclusion map i : U ↪ M {\displaystyle i:U\hookrightarrow M} , then F ( U ) {\displaystyle
Natural_bundle
the inclusion map. Suppose that F : C 0 → R {\displaystyle F:C_{0}\to \mathbb {R} } is Fréchet differentiable. Then the Fréchet derivative is a map D F
Malliavin_derivative
[X]\!]} are both coefficient rings for k {\displaystyle k} , and the inclusion map Z p → Z p [ [ X ] ] {\displaystyle \mathbb {Z} _{p}\to \mathbb {Z} _{p}[\
Deformation_ring
Topology determined by family of subspaces
recovered as the one coming from the final topology coinduced by the inclusion maps i α : C α → X α ∈ A . {\displaystyle i_{\alpha }:C_{\alpha }\to X\qquad
Coherent_topology
Abelian group
{\displaystyle b=(b_{id},b_{\iota })} where b i d {\displaystyle b_{id}} is the inclusion map and b ι {\displaystyle b_{\iota }} is sent to negative Id A {\displaystyle
Mordell–Weil_group
Most general completion of a commutative square given two morphisms with same codomain
differential maps. Preimages of sets under functions can be described as pullbacks as follows: Suppose f : A → B, B0 ⊆ B. Let g be the inclusion map B0 ↪ B
Pullback_(category_theory)
Collection of subsets that generate a topology
subspace topology, where the family consists of just one function, the inclusion map. The compact-open topology on the space of continuous functions from
Subbase
Concept in Thai historiography
the territories it subsequently lost. The maps have been widely disseminated, especially through their inclusion in Thongbai Taengnoi's student atlas, a
Territorial losses of Thailand
Territorial_losses_of_Thailand
Objects that generalize functions
of locally convex topological vector spaces and ιi : Xi → X be the inclusion maps. In this context, the inductive limit topology, or final topology, τ
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
, let i : F ~ → F S {\displaystyle i:{\tilde {F}}\to F_{S}} be the inclusion map, and define ( θ ∗ ( μ ⋅ ) ) S = i ∗ ( μ Σ ) {\displaystyle \left(\theta
Radonifying_operator
Two closely related mathematical subjects
consists of the closed points of X {\displaystyle X} with a continuous inclusion map λ X : X a n → X {\displaystyle \lambda _{X}:X^{\mathrm {an} }\to X}
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Mathematical function revertible near each point
is also locally compact, then p {\displaystyle p} is a covering map. Inclusion maps of open subsets If U ⊆ X {\displaystyle U\subseteq X} is any subspace
Local_homeomorphism
Concept in algebraic geometry
smooth complex manifold and Y be a smooth subvariety of X. Denote the inclusion map Y → X by i and the ideal sheaf of Y in X by I {\displaystyle {\mathcal
Adjunction_formula
of a free module F, and so E(M) embeds in F with the inclusion map. By composing these two maps, M is embedded in F. Anderson, Frank Wylie; Fuller, Kent
Quasi-Frobenius_ring
Algebraic structure with addition and multiplication
in S. S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism. For example, the ring Z {\displaystyle
Ring_(mathematics)
Connection on a vector bundle
H^{k}(X_{b})} where i b : X b → X {\displaystyle i_{b}\colon X_{b}\to X} is the inclusion map. Then, if we consider the classes [ i b ∗ ( ∂ i 1 + ⋯ + i n α ∂ b 1
Gauss–Manin_connection
Objects of certain abelian categories associated to topological spaces
}C)\neq 0} has real dimension at most 2i, for all i. Here jx is the inclusion map of the point x. If X is a smooth complex algebraic variety and everywhere
Perverse_sheaf
Category where every morphism is invertible; generalization of a group
_{j}:U_{ij}\to U_{j}\\t=\phi _{i}:U_{ij}\to U_{i}\end{aligned}}} and the inclusion map ε : U i → U i i {\displaystyle \varepsilon :U_{i}\to U_{ii}} giving
Groupoid
Subset of a group that forms a group itself
ab = ba = eH, then ab = ba = eG. If H is a subgroup of G, then the inclusion map H → G sending each element a of H to itself is a homomorphism. The intersection
Subgroup
Property in algebraic topology
x ) {\displaystyle \pi _{1}(U,x)\to \pi _{1}(X,x)} induced by the inclusion map of U {\displaystyle U} into X {\displaystyle X} is trivial. Here, π
Semi-locally_simply_connected
Mathematical construction
index set I {\displaystyle I} is finite. The codensity monad of the inclusion map F i n F a m ↪ F a m {\displaystyle \mathbf {FinFam} \hookrightarrow
Ultraproduct
Late Medieval map of Britain
The Gough Map or Bodleian Map is a Late Medieval map of the island of Great Britain. Its precise dates of production and authorship are unknown. It is
Gough_Map
Overview of and topical guide to discrete mathematics
targets Sign function – Function returning minus 1, zero or plus 1 Inclusion map – Set-theoretic function Pigeonhole principle – Theorem in combinatorics
Outline of discrete mathematics
Outline_of_discrete_mathematics
Vector space in mathematics
topological vector space Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous. A compatible couple (X0, X1) of Banach
Interpolation_space
Mathematical space
three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means
3-manifold
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