Suppose \(X\) is connected space and \(f\colon X\to Y\) is a continuous function . Then \(f(X)\subseteq Y\) is connected.
Proof
Without loss of generality we set \(Y=f(X)\). If \(Y\) is disconnected there are two open disjoint sets \(U,V\) such that \(Y=U\cup V\).
Then \(f^{-1}(U)\) and \(f^{-1}(V)\) disconnects \(X\).
Remarks
- This theorem implies, that a space that is homeomorphic to a connected space is itself connected.