# Example of Finite Subcovers for Explicit Open Covers of $[0,1]$

Say we have the following open cover of $$(0,1)\subset\mathbb{R}$$:

$$O=\bigcup_{n=2}^\infty \left(\frac{1}{n},1\right)$$

So this an open cover of $$(0,1)$$, but could I not turn it into an open cover of $$[0,1]$$ by defining the following set $$S$$:

$$S=O\cup (-\epsilon,\epsilon)\cup(1-\epsilon,1+\epsilon)$$

For some $$\epsilon>0$$.

Is this an open cover of $$[0,1]$$? If not why? If so how can I find a finite subcover of $$S$$ that still covers $$[0,1]$$?

Yes, $$S$$ is an open cover of $$[0,1]$$. Choose $$n\in\Bbb Z^+$$ large enough so that $$\frac1n<\epsilon$$; then
$$\{(-\epsilon,\epsilon),(1-\epsilon,1+\epsilon)\}\cup\left\{\left(\frac1k,1\right):1\le k\le n\right\}$$
is a finite subset of $$S$$ covering $$[0,1]$$.