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.
Direct Proof[edit]
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
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[edit]
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:
,
where
and
,
where
and
⋮
,
where
and
⋮
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[edit]