The compact set is defined in Topology as:
$K\subset X$ is Compact if K's every open cover has a finite subcover (ie, there exists a finite sub-collection for every open cover $\{G_\alpha\}$ that is also open cover of K).
An compact example is closed set $K=[0,1]$.
I think of three open sets: $G_1=(-0.1,0.4),G_2=(0.3,0.8), G_3=(0.7,1.2)$, then $K\subset G_1\cup G_2\cup G_3=(-0.1,1.2)$, so $\{G_\alpha\}_{\alpha=1}^3$ is an open cover of K. However, any sub-collection union of $\{G_\alpha\}_{\alpha=1}^3$ cannot cover K, which implies non-compact.
Could someone tell me where I got wrong? Thanks.
A sub-collection is not necessarily a strict sub-collection. $\{G_1,G_2,G_3\}$ is a sub-collection of $\{G_1,G_2,G_3\}$. Collection here is really synonymous with set, and a sub-collection is a subset. Every set is a subset of itself.