Answer to Question #217672 in Differential Geometry | Topology for Prathibha Rose

Question

Let X be a connected space and f:X to Y a continuous function from X onto a space Y .prove that Y is connected

Answer 1

Solution:

Proof. Suppose Y is not connected, Y = A ∪ B, where A and B are non-empty, disjoint, and open. Then $X = f^{−1} (A) ∪ f^{−1} (B)$ . These two sets are open (as preimages of open sets under a continuous function); they are disjoint (no point goes to both A and B since A and B are disjoint) and non-empty (because f is onto). Then X is not connected, a contradiction.