closure of sets closed under a finitary operation

closure of sets closed under a finitary operation

\PMlinkescapephrase

convex subset

In this entry we give a

theorem

that generalizes such results as
“the

closure

(

http://planetmath.org/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

X

be a

topological space

with a

continuous

n

-ary

operation

(

http://planetmath.org/AlgebraicSystem

)

X

n

→

X

.
If

A

⊆

X

is

closed under

this operation,
then so is

A

¯

.

Proof

Let

β

be the

n

-ary operation,
and suppose that

A

is closed under this operation,
that is,

β

⁢

(

A

×

⋯

×

A

)

⊆

A

.
From the fact that the
closure of a

product

is the product of the closures (

http://planetmath.org/ProductTopology

),
we have

β

⁢

(

A

¯

×

⋯

×

A

¯

)

=

β

⁢

(

A

×

⋯

×

A

¯

)

.

From the

characterization

of continuity in terms of closure (

http://planetmath.org/TestingForContinuityViaClosureOperation

),
we have

β

⁢

(

A

×

⋯

×

A

¯

)

⊆

β

⁢

(

A

×

⋯

×

A

)

¯

.

From the

assumption

that

β

⁢

(

A

×

⋯

×

A

)

⊆

A

,
we have

β

⁢

(

A

×

⋯

×

A

)

¯

⊆

A

¯

.

Putting all this together gives

β

⁢

(

A

¯

×

⋯

×

A

¯

)

⊆

A

¯

,

as required.

Examples

If

H

is a subgroup of a

topological group

G

,
then

H

is closed under both the

group operation

and the operation of

inversion

,
both of which are continuous,
and therefore by the theorem

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

↦

g

-

1

⁢

x

⁢

g

for each

g

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

↦

r

⁢

x

for each

r

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

λ

the unary operation

x

↦

λ

⁢

x

.

As a final example, we look at convex sets.
Let

A

be a convex subset of a real (or complex) topological vector space.
Convexity means that for every

t

∈

[

0

,

1

]

the set is closed under the

binary operation

(

x

,

y

)

↦

(

1

-

t

)

⁢

x

+

t

⁢

y

.
These binary operations are all continuous,
so the theorem again applies, and we conclude that

A

¯

is convex.

Title

closure of sets closed under a finitary operation

Canonical name

ClosureOfSetsClosedUnderAFinitaryOperation

Date of creation

2013-03-22 17:03:29

Last modified on

2013-03-22 17:03:29

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

20

Author

yark (2760)

Entry type

Theorem

Classification

msc 52A07

Classification

msc 57N17

Classification

msc 13J99

Classification

msc 22A05

Related topic

ClosureOfAVectorSubspaceIsAVectorSubspace

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ClosureOfAVectorSubspaceIsAVectorSubspace2

Related topic

FreelyGeneratedInductiveSet

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