Inductive/final/direct limit topology

Throughout, it is assumed that

• ${\displaystyle {\mathcal {C}}}$ is either the category of topological spaces or some subcategory of the category of topological vector spaces (TVSs);
  • If all objects in the category have an algebraic structure, then all morphisms are assumed to be homomorphisms for that algebraic structure.
• I is a non-empty directed set;
• X_• = ( X_i )_{i ∈ I} is a family of objects in ${\displaystyle {\mathcal {C}}}$ where (X_i, τ_Xi) is a topological space for every index i;
  • To avoid potential confusion, τ_Xi should not be called X_i's "initial topology" since the term "initial topology" already has a well-known definition. The topology τ_Xi is called the original topology on X_i or X_i's given topology.
• X is a set (and if objects in ${\displaystyle {\mathcal {C}}}$ also have algebraic structures, then X is automatically assumed to have has whatever algebraic structure is needed);
• f_• = ( f_i )_{i ∈ I} is a family of maps where for each index i, the map has prototype f_i : (X_i, τ_Xi) → X. If all objects in the category have an algebraic structure, then these maps are also assumed to be homomorphisms for that algebraic structure.

If it exists, then the final topology on X in ${\displaystyle {\mathcal {C}}}$, also called the colimit or inductive topology in ${\displaystyle {\mathcal {C}}}$, and denoted by τ_f• or τ_f, is the finest topology on X such that

1. (X, τ_f) is an object in ${\displaystyle {\mathcal {C}}}$, and
2. for every index i, the map f_i : (X_i, τ_Xi) → (X, τ_f) is a continuous morphism in ${\displaystyle {\mathcal {C}}}$.

In the category of topological spaces, the final topology always exists and moreover, a subset U ⊆ X is open (resp. closed) in (X, τ_f) if and only if f_i^- 1 (U) is open (resp. closed) in (X_i, τ_Xi) for every index i.

However, the final topology may not exist in the category of Hausdorff topological spaces due to the requirement that (X, τ_Xf) belong to the original category (i.e. belong to the category of Hausdorff topological spaces).^[3]

Direct systems

Suppose that (I, ≤) is a directed set and that for all indices i ≤ j there are (continuous) morphisms in ${\displaystyle {\mathcal {C}}}$

such that if i = j then f_i^j is the identity map on X_i and if i ≤ j ≤ k then the following compatibility condition is satisfied:

where this means that the composition

${\displaystyle X_{i}\xrightarrow {f_{i}^{j}} X_{j}\xrightarrow {f_{j}^{k}} X_{k}\;\;\;\;{\text{ is equal to }}\;\;\;\;X_{i}\xrightarrow {f_{i}^{k}} X_{k}.}$

If the above conditions are satisfied then the triple formed by the collections of these objects, morphisms, and the indexing set

${\displaystyle \left(X_{\bullet },\left\{f_{i}^{j}\;:\;i,j\in I\;{\text{ and }}\;i\leq j\right\},I\right)}$

is known as a direct system in the category ${\displaystyle {\mathcal {C}}}$ that is directed (or indexed) by I. Since the indexing set I is a directed set, the direct system is said to be directed.^[4] The maps f_i^j are called the bonding, connecting, or linking maps of the system.

If the indexing set I is understood then I is often omitted from the above tuple (i.e. not written); the same is true for the bonding maps if they are understood. Consequently, one often sees written "X_• is a direct system" where "X_•" actually represents a triple with the bonding maps and indexing set either defined elsewhere (e.g. canonical bonding maps, such as natural inclusions) or else the bonding maps are merely assumed to exist but there is no need to assign symbols to them (e.g. the bonding maps are not needed to state a theorem).

Direct limit of a direct system

For the construction of a direct limit of a general inductive system, please see the article: direct limit.

Direct limits of injective systems

If each of the bonding maps ${\displaystyle f_{i}^{j}}$ is injective then the system is called injective.^[4]

Assumptions: In the case where the direct system is injective, it is often assumed without loss of generality that for all indices i ≤ j, each X_i is a vector subspace of X_j (in particular, X_i is identified with the range of ${\displaystyle f_{i}^{j}}$) and that the bonding map ${\displaystyle f_{i}^{j}}$ is the natural inclusion

In^j
_i : X_i → X_j

(i.e. defined by x ↦ x) so that the subspace topology on X_i induced by X_j is weaker (i.e. coarser) than the original (i.e. given) topology on X_i.

In this case, also take

X := ∪i ∈ I X_i.

The limit maps are then the natural inclusions In_i : X_i → X. The direct limit topology on X is the final topology induced by these inclusion maps.

If the X_i's have an algebraic structure, say addition for example, then for any x, y ∈ X, we pick any index i such that x, y ∈ X_i and then define their sum using by using the addition operator of X_i. That is,

where +_i is the addition operator of X_i. This sum is independent of the index i that is chosen.

In the category of locally convex topological vector spaces, the topology on the direct limit X of an injective directed inductive limit of locally convex spaces can be described by specifying that an absolutely convex subset U of X is a neighborhood of 0 if and only if U ∩ X_i is an absolutely convex neighborhood of 0 in X_i for every index i.^[4]

Direct limits in Top

Direct limits of directed direct systems always exist in the categories of sets, topological spaces, groups, and locally convex TVSs. In the category of topological spaces, if every bonding map f_i^j is/is a injective (resp. surjective, bijective, homeomorphism, topological embedding, quotient map) then so is every f_i : X_i → X.^[3]

Strict inductive limit

If each of the bonding maps ${\displaystyle f_{i}^{j}}$ is an embedding of TVSs onto proper vector subspaces and if the system is directed by ${\displaystyle \mathbb {N} }$ with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each X_i is a vector subspace of X_i+1 and that the subspace topology induced on X_i by X_i+1 is identical to the original topology on X_i.^[1]

In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces X can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if U ∩ X_n is an absolutely convex neighborhood of 0 in X_n for every n.

LF-spaces

Every LF-space is a meager subset of itself.^[6] The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete. In particular, every LF-space is complete.^[7] Every LF-space is barrelled and bornological, which together with completeness implies that every LF-space is ultrabornological. An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.^[8] LF spaces are distinguished and their strong duals are bornological and barrelled (a result due to Alexander Grothendieck).

If X is the strict inductive limit of an increasing sequence of Fréchet space X_n then a subset B of X is bounded in X if and only if there exists some n such that B is a bounded subset of X_n.^[7]

A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.^[9] A linear map from an LF-space X into a Fréchet space Y is continuous if and only if its graph is closed in X × Y.^[10] Every bounded linear operator from an LF-space into another TVS is continuous.^[11]

If X is an LF-space defined by a sequence ${\displaystyle \left(X_{i}\right)_{i=1}^{\infty }}$ then the strong dual space ${\displaystyle X_{b}^{\prime }}$ of X is a Fréchet space if and only if all X_i are normable.^[12] Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.

Space of smooth compactly supported functions

A typical example of an LF-space is, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$, the space of all infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support. The LF-space structure is obtained by considering a sequence of compact sets ${\displaystyle K_{1}\subset K_{2}\subset \ldots \subset K_{i}\subset \ldots \subset \mathbb {R} ^{n}}$ with ${\displaystyle \bigcup _{i}K_{i}=\mathbb {R} ^{n}}$ and for all i, ${\displaystyle K_{i}}$ is a subset of the interior of ${\displaystyle K_{i+1}}$. Such a sequence could be the balls of radius i centered at the origin. The space ${\displaystyle C_{c}^{\infty }(K_{i})}$ of infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support contained in ${\displaystyle K_{i}}$ has a natural Fréchet space structure and ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets ${\displaystyle K_{i}}$.

With this LF-space structure, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ is known as the space of test functions, of fundamental importance in the theory of distributions.

Direct limit of finite-dimensional spaces

Suppose that for every positive integer n, X_n := ${\displaystyle \mathbb {R} }$ⁿ and for m < n, consider X_m as a vector subspace of X_n via the canonical embedding X_m → X_n defined by x := (x₁, ..., x_m) ↦ (x₁, ..., x_m, 0, ..., 0). Denote the resulting LF-space by X. Since any TVS topology on X makes continuous the inclusions of the X_m's into X, the latter space has the maximum among all TVS topologies on an ${\displaystyle \mathbb {R} }$-vector space with countable Hamel dimension. It is a LC topology, associated with the family of all seminorms on X. Also, the TVS inductive limit topology of X coincides with the topological inductive limit; that is, the direct limit of the finite dimensional spaces X_n in the category TOP and in the category TVS coincide. The continuous dual space ${\displaystyle X^{\prime }}$ of X is equal to the algebraic dual space of X, that is the space of all real valued sequences ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ and the weak topology on ${\displaystyle X^{\prime }}$ is equal to the strong topology on ${\displaystyle X^{\prime }}$ (i.e. ${\displaystyle X_{\sigma }^{\prime }=X_{b}^{\prime }}$).^[13]. In fact, it is the unique LC topology on ${\displaystyle X^{\prime }}$ whose topological dual space is X. Furthermore, the canonical map of X into the continuous dual space of ${\displaystyle X_{\sigma }^{\prime }}$ is surjective.^[13]