Problem 2

Exercise 8.2 (Continuous maps preserve connectedness)   If $X$ is connected and $f : X \to Y$ is continuous, then $f(X)$ is connected.

Proof. Suppose $f(X)=A\cup B$ is a separation. Then $X$ has a separation:

$$X = f^{-1}(f(X)) = f^{-1}(A\cup B) = f^{-1}(A)\cup f^{-1}(B).$$ (8.4)

This contradicts that $X$ is connected, so we must instead have that $f(X)$ is connected. $\qedsymbol$