\b[True.] A separation of $Y$ into two disjoint nonempty open subsets would pull back under the continuous <surjection> $f$ to a separation of $X$. The <inverse images> are nonempty by <surjectivity>, contradicting <connectedness> of $X$. Thus a <continuous image of a connected space> is connected.

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / E / 3 / b / Solution

 Solution (source code)