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Pointwise limits of homomorphisms and of equicontinuous characters

Statement

(1) Let $G$ be a group, written additively. A pointwise limit of group homomorphisms $G \to T$ is a homomorphism; precisely, $Hom (G, T)$ is closed in $T^{G}$ for the topology of pointwise convergence (The topology of pointwise convergence on $Y^{X}$ , which is the product topology, and its restriction to $C (X, Y)$ , The product set $\prod_{i \in I} X_{i}$ 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). (2) For an abelian topological group $G$ , if a pointwise limit of continuous homomorphisms is taken along an equicontinuous family, then the limit is continuous, hence a character. (3) If the abelian topological group $G$ is discrete (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies), then $\hat{G} = Hom (G, T)$ is closed in $T^{G}$ for the product topology.

Facts & Assumptions

[F1]

Multiplication in $T$ is continuous. $T^{G}$ is the product of the constant family with one factor $T$ per $x \in G$ ; a map into a product is continuous exactly when all its components are, and the projection $f \mapsto f (x)$ is continuous for every $x$ . (The product set $\prod_{i \in I} X_{i}$ 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 multiplicative unit circle is a compact metrizable topological abelian group, Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous)

[F2]

$T$ is Hausdorff, and the diagonal $Δ = {(z, z) : z \in T}$ is closed in $T \times T$ : if $z \neq w$ , disjoint open neighbourhoods of $z$ and $w$ give an open rectangle around $(z, w)$ missing $Δ$ . (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not, Distinct points of a metric space have disjoint balls around them, The product set $\prod_{i \in I} X_{i}$ 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)

[F3]

A point lies in the closure of a set exactly when some net in the set converges to it. (A point lies in the closure of a set if and only if a net in the set converges to it, Directed preorders and nets)

[F4]

The closure in $Y^{X}$ with the topology of pointwise convergence of an equicontinuous family $F \subseteq C (X, Y)$ into a metric space $Y$ is equicontinuous, and every member of that closure is continuous; the topology of pointwise convergence on $C (X, Y)$ is the subspace topology inherited from $Y^{X}$ . (The pointwise closure of an equicontinuous family is equicontinuous and consists of continuous maps, Equicontinuity on a topological domain and pointwise relative compactness, The topology of pointwise convergence on $Y^{X}$ , which is the product topology, and its restriction to $C (X, Y)$ )

[F5]

A discrete topology makes every subset open, and continuity means that for each point and open neighbourhood of its image there is an open source neighbourhood mapped into it. (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Continuity of a map of topological spaces at a point and globally)

[F6]

The dual consists of the continuous homomorphisms $G \to T$ with the compact-open topology, and the compact-open topology on a discrete domain agrees with the topology of pointwise convergence, i.e. with the subspace topology from $T^{G}$ . (The Pontryagin dual with the compact-open topology, On a discrete domain the compact-open topology is the topology of pointwise convergence)

Proof

Given: A group $G$ , the product space $T^{G}$ with the topology of pointwise convergence, and the set $Hom (G, T)$ of all group homomorphisms $G \to T$ .

1.1F1F2

$Hom (G, T)$ is closed in $T^{G}$ : it is the intersection over all $x, y \in G$ of the sets $E_{x, y} : = {f : f (x + y) = f (x) f (y)}$ , and each $E_{x, y}$ is the preimage of the diagonal $Δ \subseteq T \times T$ under the map $φ_{x, y} (f) : = (f (x + y), f (x) f (y))$ , which is continuous because both components are continuous by [F1]; preimages of the closed set $Δ$ under continuous maps are closed by [F2], and arbitrary intersections of closed sets are closed.

2.1step 1.1F3

Consequently a pointwise limit of group homomorphisms is a homomorphism: if a net $(γ_{j})$ in $Hom (G, T)$ converges pointwise to $γ$ , then $γ$ lies in the closure of $Hom (G, T)$ by [F3], and that closure equals the closed set $Hom (G, T)$ by step 1.1, so $γ$ is a homomorphism.

2.2step 1.1F5F6

If $G$ is discrete, then for any map $f : G \to T$ , point $x \in G$ and open set $V$ containing $f (x)$ , the preimage $f^{- 1} [V]$ is a subset of $G$ , hence open and contains $x$ ; it maps into $V$ , so the continuity definition [F5] makes $f$ continuous at every $x$ . Thus every map $G \to T$ is continuous, so the continuous characters are exactly the homomorphisms, $\hat{G} = Hom (G, T)$ , and this set is closed in $T^{G}$ by step 1.1; by [F6] the compact-open topology on $\hat{G}$ is its subspace topology from $T^{G}$ , so $\hat{G}$ is a closed subset of $T^{G}$ as asserted.

3.1step 2.1F4

If the homomorphisms $γ_{j}$ above are continuous and the family ${γ_{j}}$ is equicontinuous, then $γ$ is continuous: the family lies in $C (G, T)$ , its pointwise closure is equicontinuous and consists of continuous functions by [F4], and $γ$ belongs to that closure.

4.1step 2.1step 3.1step 2.2∎

Clauses (1), (2) and (3) of the statement are steps 2.1, 3.1 and 2.2 respectively.

Depends on

Used by

Dependency tree · two levels

83 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Dikran D. Dikranjan, Introduction to Topological Groups (author lecture notes, Universita di Udine / Universidad Complutense de Madrid, 2007) (standard reference, not scraped)
Lynn H. Loomis, Introduction to Abstract Harmonic Analysis, D. Van Nostrand 1953, Chapter VII, Sections 34-35 (printed pp. 134-140) (standard reference, not scraped)