In this post on Reddit, a user proposes an alternate definition of compactness, as an "induction principle":
A topological space $X$ is compact iff given a statement $P$ whose truth or falsity is defined on open subsets of $X$, we have $P(X)$ as soon as
For all $x \in X$ there exists open $U \ni x$ such that $P(U)$.
$P(U_1), \dotsc, P(U_n) \implies P(U_1 \cup \dotsb \cup U_n)$.
To me this seems like such a drastically better definition of compactness: more intuitive, more mathematical, more to the point. The statement in terms of open covers is useful in proofs, but it should be secondary to this.
Any thoughts/opinions? Why don't we teach compactness this way, except that that's the way it's always been done?