EDB — 0PR

[0PR] Let \(E⊆ X\), then \(E\) is a metric space with the restricted distance \(\tilde d=d|_{E× E}\).

Show that \(A⊆ E\) is open in \((E,\tilde d)\) (as defined at the beginning of this section) if and only there exists a set \(V⊆ X\) open in \((X,d)\) such that \(V∩ E = A\).

(The second way of defining ”open” is used in topology.)