Given a cover of a compact metric space, all small subsets are subset of some cover set
In topology, the Delta number, is a useful tool in the study of compact metric spaces. It states:
- If the metric space
is compact and an open cover of
is given, then there exists a number
such that every subset of
having diameter less than
is contained in some member of the cover.
Such a number
is called a Delta number of this cover. The notion of a Delta number itself is useful in other applications as well.
Proof
Direct Proof
Let
be an open cover of
. Since
is compact we can extract a finite subcover
.
If any one of the
's equals
then any
will serve as a Delta number.
Otherwise for each
, let
, note that
is not empty, and define a function
by
![{\displaystyle f(x):={\frac {1}{n}}\sum _{i=1}^{n}d(x,C_{i}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/650de14149f820ac39df9cf89d8c194a54ac5b3d)
Since
is continuous on a compact set, it attains a minimum
.
The key observation is that, since every
is contained in some
, the extreme value theorem shows
. Now we can verify that this
is the desired Delta number.
If
is a subset of
of diameter less than
, choose
as any point in
, then by definition of diameter,
, where
denotes the ball of radius
centered at
. Since
there must exist at least one
such that
. But this means that
and so, in particular,
.
Proof by Contradiction
Assume
is sequentially compact,
is an open covering of
and the Lebesgue number
does not exist. So,
,
with
such that
where
.
This allows us to make the following construction:
![{\displaystyle \delta _{1}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7728373cc7bf6d6436aab5ad75de23b841773536)
,
![{\displaystyle \exists A_{1}\subset X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8ac147bb79a4f4c63c7ac32b5b4da9770f21f6)
where
![{\displaystyle (diam(A_{1})<\delta _{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95663d0e960e6954a9d7a08ff1a09558a5008391)
and
![{\displaystyle \neg \exists \beta (A_{1}\subset U_{\beta })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/528dbff442fe0d8f35c9c9b2ffbcc201fa6aed0d)
![{\displaystyle \delta _{2}={\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8280eb36d128133d0295c63ce566bab6be3acea)
,
![{\displaystyle \exists A_{2}\subset X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23e866140840f94f9a7dae5f4ec5c756c3e08248)
where
![{\displaystyle (diam(A_{2})<\delta _{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d9ee672b6a9701e6b96dedad5bdac53ddd238b)
and
![{\displaystyle \neg \exists \beta (A_{2}\subset U_{\beta })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa00ed8a205ce895b22d6b801fd451eb6fd347d)
⋮
![{\displaystyle \delta _{k}={\frac {1}{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38130b54b82d34910cbc6482fc32b9171a08dffb)
,
![{\displaystyle \exists A_{k}\subset X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7869b99b17fc2a2f45199bea9d4b26423749458)
where
![{\displaystyle (diam(A_{k})<\delta _{k})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/407f68a9c31afe8c72dd4fab0b472bbd46c8bd4e)
and
![{\displaystyle \neg \exists \beta (A_{k}\subset U_{\beta })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d840b91fb1dd3981edafc1e258136f347949e1)
⋮
For all
,
since
.
It is therefore possible to generate a sequence
where
by axiom of choice. By sequential compactness, there exists a subsequence
that converges to
.
Using the fact that
is an open covering,
where
. As
is open,
such that
. By definition of convergence,
such that
for all
.
Furthermore,
where
. So,
.
Finally, let
such that
and
. For all
, notice that:
because
.
because
which means
.
By the triangle inequality,
, implying that
which is a contradiction.
References