In Basic Topology, Armstrong states, "a topological space is compact if every open cover of X has a finite subcover." But, I'm mildly confused how this doesn't translate to the empty set being a finite subcover itself.
First, let X be a compact space and let U be an open cover of X. Thus, there exists a finite subcover of X, say $\{x_1, ..., x_n\}$. But then, this itself is an open cover of X, since a set of open sets whose union is the space is an open cover. Thus, there exists some element $x_i$ s.t. $$\bigcup\{x_1, \ldots, \widehat{x_i},\ldots x_n\}= X$$ i.e., a set without $x_i$ in it that's an open cover of X.
Why couldn't we repeat this until the set is empty and then we get $$\bigcup \emptyset = X$$ which is obviously a contradiction.
I obviously have some deep and fundamental misunderstanding of what open covers, subcovers, and compact spaces are, so I'd appreciate any insight.