Why is the affine $1$-space $\mathbb{A}^1$ non-compact? A topological space $X$ is called compact if every open cover has a finite (open) subcover, by definition. Why is the affine $1$-space $\mathbb{A}^1$ considered non-compact, in the topology used in algebraic geometry, where the zero-locus of polynomials in $1$-indeterminates are the closed sets and we do not have the Hausdorff property. I know that irreducible closed sets are all singletons and all other closed sets, except the whole space $\mathbb{A}^1$, are finite sets. Every open set equals the whole of $\mathbb{A}^1$ with a finite number of points removed. I am not sure how I can construct an open cover of $\mathbb{A}^n$ that has no finite (open) subcover?
 A: Every affine variety is quasi-compact in the sense that every open cover has a finite subcover. This boils down to a fact from algebra: If $(I_s)_{s \in S}$ is a family of ideals of a ring $R$ with $\sum_{s \in S} I_s = R$, then $\sum_{s \in T} I_s = R$ for a finite subset $T \subseteq S$.
Most authors define compact := quasi-compact (i.e., the term "quasi-compact" is not necessary at all), but some authors (in particular within the Bourbaki school) define compact := quasi-compact + Hausdorff (see also Wikipedia). In this sense, only zero-dimensional varieties are compact, and in particular $\mathbb{A}^1$ would be non-compact (but quasi-compact).
In my opinion, overloading such a central notion of topology should be avoided at all costs. Also, there is no reason why the Hausdorff property should be included in the notion of a compact space, since the word "compact" refers to a kind of smallness, and Hausdorffness has nothing to do with smallness – it is a separation property. So I would suggest that we all agree on the definition compact := every open cover has a finite subcover. This has the benefit that no reader has to wonder if an author talking about compact spaces assumes Hausdorffness or not. It would also eliminate the need for an extra term like "quasi-compact". (Unfortunately, this will probably not happen, since the terminologies of quasi-compact schemes and morphisms are already too much engrained in the algebraic geometry community and literature.) And when you need both properties, you can just say "compact Hausdorff", as most authors already do.
There is a similar discussion for settling the notion of a ring, see Poonen's Why all rings should have a 1.
