Let $K$ be a compact subset of a metric space, and I cover $K$ with finite open sets. How can I select $\delta>0$, such that for any $x,y\in K$, with $d(x,y) <\delta$, $x,y$ are inside an open set?
I get this question when I try to show for any compact subset $K$ of a Riemannian manifold $M$, there exists $\epsilon>0$ s.t.$\forall y,z\in K$ with $d(y,z)<\epsilon$, then there exists a minimizing geodesic joining $y,z$. My approach is covering $K$ by totally normal neighborhood $W_\alpha$ of finitely many points in $K$, and what I have to do is to select small enough $\epsilon$, such that any two points with distance less that $\epsilon$ are inside a common totally normal neighborhood.
I think this can be done since I only need to make sure the points are sufficiently far away from the boundary. Could you help me to find a rigorous argument? Thanks.