Question

Let {Uα } α be an open cover of a compact set K in a metric space (X,d). Show that there is ϵ>0, so that if p,q∈K:rho(p,q)<ϵ, then there exists α=α(p,q,ϵ), so that p,q∈Uα

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Answer 1

The question pertains to showing the existence of an ε>0 for points in a compact set K within a metric space that are covered by the same open set of an open cover. By utilizing the compactness property and choosing a finite subcover, we can find an ε that satisfies the property for any two points within K.

Step-by-step explanation:

The question is about a concept in topology, specifically dealing with compact sets and open covers in a metric space. To show that there exists an ε>0 such that for any p, q ∈ K with d(p,q) < ε, there is an open set Uα containing both p and q, we start by considering the compactness of K.

For each point x ∈ K, there is an open set Uαx containing x. Because K is compact, there must be a finite subcover of the open cover of K, say {Uα1, Uα2, ..., Uαn}. For each x ∈ K, choose εx > 0 such that the ball B(x, εx) (ε-neighborhood) is contained in Uαx. The minimum of these εxs (which is positive since we have a finite number of them) gives the desired ε for our proof.

Now, if d(p,q) < ε, both p and q must lie within some ε-ball contained in one of the finite subcovers, meaning they are contained in the same Uα. This concludes that there is an ε>0 which satisfies the property stated in the question.