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closure of sets closed under a finitary operation

(Theorem)

In this entry we give a theorem that generalizes such results as “the closure of a subgroup is a subgroup” and “the closure of a convex set is convex”.

Theorem and proof

Since the theorem involves two different concepts of closure -- algebraic and topological -- we must be careful how we phrase it.

Theorem Let be a topological space with a continuous -ary operation $X^n\to X$ . If $A\subseteq X$ is closed under this operation, then so is $\overline {A}$ .

Proof

Let $\beta$ be the -ary operation, and suppose that is closed under this operation, that is, $\beta(A\times\cdots\times A)\subseteq A$ . From the fact that the closure of a product is the product of the closures, we have

$\begin{displaymath} \beta(\overline {A}\times\cdots\times\overline {A}) = \beta(\overline {A\times\cdots\times A}). \end{displaymath}$

From the characterization of continuity in terms of closure, we have

$\begin{displaymath} \beta(\overline {A\times\cdots\times A}) \subseteq \overline {\beta(A\times\cdots\times A)}. \end{displaymath}$

From that assumption that $\beta(A\times\cdots\times A)\subseteq A$ , we have

$\begin{displaymath} \overline {\beta(A\times\cdots\times A)} \subseteq \overline {A}. \end{displaymath}$

Putting all this together gives

$\begin{displaymath} \beta(\overline {A}\times\cdots\times\overline {A}) \subseteq \overline {A}, \end{displaymath}$

as required.

Examples

If is a subgroup of a topological group , then is closed under both the group operation and the operation of inversion, both of which are continuous, and therefore by the theorem $\overline {H}$ is also closed under both operations. Thus the closure of a subgroup of a topological group is also a subgroup.

It similarly follows that the closure of a normal subgroup of a topological group is a normal subgroup. In this case there are additional unary operations to consider: the maps $x\mapsto g^{-1}xg$ for each in the group. But these maps are all continuous, so the theorem again applies.

Note that it does not follow that the closure of a characteristic subgroup of a topological group is characteristic, because this would require applying the theorem to arbitrary automorphisms of the group, and these automorphisms need not be continuous.

Straightforward application of the theorem also shows that the closure of a subring of a topological ring is a subring. Considering also the unary operations $x\mapsto rx$ for each in the ring, we see that the closure of a left ideal of a topological ring is a left ideal. Similarly, the closure of a right ideal of a topological ring is a right ideal.

We also see that the closure of a vector subspace of a topological vector space is a vector subspace. In this case the operations to consider are vector addition and for each scalar $\lambda$ the unary operation $x\mapsto\lambda x$ .

As a final example, we look at convex sets. Let be a convex subset of a real (or complex) topological vector space. Convexity means that for every $t\in[0,1]$ the set is closed under the binary operation $(x,y)\mapsto(1-t)x+ty$ . These binary operations are all continuous, so the theorem again applies, and we conclude that $\overline {A}$ is convex.

"closure of sets closed under a finitary operation" is owned by yark.

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Cross-references: binary operation, complex, real, convex sets, scalar, vector addition, topological vector space, vector subspace, right ideal, left ideal, ring, topological ring, subring, automorphisms, characteristic, characteristic subgroup, group, maps, unary, normal subgroup, group operation, topological group, subgroup, closed under, continuous, topological space
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This is version 15 of closure of sets closed under a finitary operation, born on 2007-05-09, modified 2007-05-11.
Object id is 9349, canonical name is ClosureOfSetsClosedUnderAFinitaryOperation.
Accessed 999 times total.

Classification:

AMS MSC:	22A05 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Structure of general topological groups)
	13J99 (Commutative rings and algebras :: Topological rings and modules :: Miscellaneous)
	57N17 (Manifolds and cell complexes :: Topological manifolds :: Topology of topological vector spaces)
	52A07 (Convex and discrete geometry :: General convexity :: Convex sets in topological vector spaces)

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