I've heard during a lecture that a "set defined by equations is always closed". The equation was a matricial equation of the type $$AA^T=I$$ The lecturer didn't spend more than this sentence on the topic so either is trivial or either is false.
Are there conditions on the cardinality of equations which turn the proposition true or false?
Well, suppose that by equation we mean an expression of the form $f(x) \ge y$, $f(x) \le y$ or $f(x) = y$, where $f$ is some continuous function on the space to the real numbers (for concreteness). Then all such sets are closed, as the inverse image of the continuous $f$ of the closed sets $(-\infty, y], [y, +\infty), \{y\}$ of the reals. And it doesn't matter how many equations we have (for different $f$ or different $y$ etc.), because then we have an intersection of closed sets, and any intersection of closed sets is closed.