TITLE: Unnecessary uses of the Continuum Hypothesis
QUESTION [19 upvotes]: This question was inspired by the MathOverflow question "Unnecessary uses of the axiom of choice". I want to know of statements in ZFC that can be proven by assuming the Continuum Hypothesis, but can also be proven by a more elaborate (perhaps even significantly more elaborate) proof without assuming the Continuum Hypothesis.

REPLY [7 votes]: A Dowker space is a normal Hausdorff space whose product with
the closed unit interval $I$ is not normal. In 1971, Mary Ellen Rudin constructed the first ZFC Dowker space, which had cardinality $\aleph_\omega^\omega$. This space was considered "large" and for a long time it was an open problem to construct a "small" Dowker space.  Various people constructed small Dowker spaces by assuming extra hypotheses.  Most relevant to the current question is the construction, by I. Juhász, K. Kunen and M. E. Rudin (Two more hereditarily separable non-Lindelöf spaces), of a small Dowker space assuming CH.  Finally, in 1996, Zoltán Balogh  constructed A small Dowker space in ZFC.
There might be other examples stemming from the work of M. E. Rudin.  Initially, I thought that the normality of ${\mathbb R}^\omega$ in the box topology might be an answer to this question, but apparently it is still open whether it can be proved without CH.