continuous map and continuity of the restrictions of an open covering

1. Proposition

Let \((X,\mathcal{T})\) and \((X',\mathcal{T}')\) be topological spaces, \(\bigcup_{i \in I} O_i = X\) an open covering of \(X\) and \(f: X \rightarrow X'\) a map. Then the following statements are equivalent

\(f: X \rightarrow X'\) is continuous
each restriction \(f_{O_i}: O_i \rightarrow X'\) is continuous

2. Proof

2.1. a)

Let \(O' \subseteq X'\) be an open set. Then \(O := f^{-1}[O']\) is open. Therefore it follows from the Openness of a finite intersection of open sets, that \(O \cap O_i\) is open

2.2. b)

follows from the glueing lemma for an open covering