Open Cover

See also Finite Subcover.

Example

Consider the open interval $(0,1)$. For each point $x\in (0,1)$ let $O_x=(\frac{x}{2},1)$. Taken together $\{O_x:x\in (0,1)\}$ forms an open cover for the interval $(0,1)$. Notice that it is impossible to find a Finite Subcover. Given any proposed finite subcollection:

then let $x^{\prime }=min\{x_1,x_2,\dots ,x_n\}$ and observe that we can construct a number $y$ satisfying $0<y\le \frac{x^{\prime }}{2}$. Then $y\notin \bigcup _{i=1}^{\mathrm{\infty }}{O}_{x_i}$.

Now consider a similar subcover for the closed interval $[0,1]$. For $x\in (0,1)$ the set $O_x$ defined above works covering $(0,1)$ but in order to have an open cover of the closed $[0,1]$ interval then we need the endpoints. Let $\epsilon >0$ and let $O_0=(-\epsilon ,\epsilon )$ and $O_1=(1-\epsilon ,1+\epsilon )$. Then the collection:

is an open cover for $[0,1]$. But this time notice that there is a finite subcover. This is because the addition of $O_0$ is here, we can choose $x^{\prime }$ so that $\frac{x^{\prime }}{2}<\epsilon$. It follows that our collection