# closure of sets closed under a finitary operation

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) ${X}^{n}\mathrm{\to}X$.
If $A\mathrm{\subseteq}X$ is closed under this operation,
then so is $\overline{A}$.

Proof

Let $\beta $ be the $n$-ary operation,
and suppose that $A$ is closed under this operation,
that is, $\beta (A\times \mathrm{\cdots}\times A)\subseteq A$.
From the fact that the
closure of a product^{} is the product of the closures (http://planetmath.org/ProductTopology),
we have

$$\beta (\overline{A}\times \mathrm{\cdots}\times \overline{A})=\beta (\overline{A\times \mathrm{\cdots}\times A}).$$ |

From the characterization of continuity in terms of closure (http://planetmath.org/TestingForContinuityViaClosureOperation), we have

$$\beta (\overline{A\times \mathrm{\cdots}\times A})\subseteq \overline{\beta (A\times \mathrm{\cdots}\times A)}.$$ |

From the assumption^{} that $\beta (A\times \mathrm{\cdots}\times A)\subseteq A$,
we have

$$\overline{\beta (A\times \mathrm{\cdots}\times A)}\subseteq \overline{A}.$$ |

Putting all this together gives

$$\beta (\overline{A}\times \mathrm{\cdots}\times \overline{A})\subseteq \overline{A},$$ |

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 $\overline{H}$
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 $g$ 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 $x\mapsto rx$ 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 $\lambda $ the unary operation $x\mapsto \lambda x$.

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 $t\in [0,1]$
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.

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 |

Related topic | ClosureOfAVectorSubspaceIsAVectorSubspace2 |

Related topic | FreelyGeneratedInductiveSet |