Continuing horrors of topology without choice (Q1892171)

The authors show that many familiar topological results require some form of the axiom of choice in an essential way. Among the more unnerving results are that, without the axiom of choice, (1) \(\omega_ 1\) may be paracompact and even Lindelöf; (2) there may be a compact metric space which is neither separable nor second-countable; and (3) there may be a Suslin line whose square is ccc.