In Mathematics (and in related disciplines, like Theoretical Computer Science), an open question asks to prove something is true, or alternatively to prove that it is false. Mathematicians accept proof as a measure of truth, so once a satisfactory proof is found the question is considered "closed".

Usually most people believe that some particular resolution to the question will be shown. For instance, the majority of people in CS believe P!=NP (see below). But lacking proof this has no merit.

Sometimes a different type of resolution is found. As an example, the Continuum Hypothesis was first posed as a statement to be proved -- an open question. But the work of Paul Cohen (along with earlier work by Kurt Gödel) showed that neither the statement nor its negation can be proved in ZFC (assuming consistency of ZFC). The question was closed, but not in a way Georg Cantor would have imagined possible.

The top open question in mathematics today is the Riemann Hypothesis. The top open question in CS today is P=NP? / P!=NP (all the same question).