Checking compactness of $[0,1]$ using the definition

Question

By definition a set is compact if every open covering has a finite subcover. I made an open covering of $[0,1]$ by taking a fixed $\epsilon= \dfrac1{10,00000}$ (or even more small) radius neighborhood around each point in $[0,1$]. But I could not find its finite subcover. By Heine-Borel theorem it's clear that it will be compact. But, I was trying to use definition but could not find its finite subcover.

Answer 1

The size of $\varepsilon$ doesn't matter. Take the intervals $(-\varepsilon,\varepsilon)$, $(-\varepsilon/2,3\varepsilon/2)$, $(0,2\varepsilon)$, and so on…

Answer 2

Hi OK this is a good fundamental question about compactness.

If you take neighbourhoods of width $1/1,000,000$ around $0$,$1/1,000,000$,$2/1,000,000$,$3/1,000,000$,$4/1,000,000$,$\dots$, $999,999/1,000,000$ and $1$ then this covers the space and is finite (even though it is very big). As epsilon gets smaller then the number of sets that is required increases (in inverse proportion to epsilon) but it will always be finite.