Let \((X,d)\) be a metric space. A subset \(U\subseteq X\) is open, if for each \(x\in U\) there exists \(\epsilon > 0\) such that the open ball is contained in \(U\): \[B(x,\epsilon)\subset U,\] i.e. if \(U\) it is the neighborhood of any of its points. Please note that, by definition, every open ball \(B(x, r)\) is open.
Definitions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Lemmas: 15
Parts: 16 17
Proofs: 18 19 20 21 22 23 24 25 26
Propositions: 27 28 29
Theorems: 30