continuous map and continuity of the restrictions of a locally finite, closed covering

1. Proposition

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

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

2. Proof

2.1. a)

Let \(A' \subseteq X'\) be a closed set. Then \(A := f^{-1}[A']\) is closed. Therefore, since the intersection of closed sets is closed that \(A \cap A_i\) is closed

2.2. b)

follows from the glueing lemma for a locally finite, closed covering