A subset H of a Euclidean space X is said to be compact (in X) if every open covering of H has a finite subcovering.Quote:

Originally Posted byPanconleche

Here are a few examples.

Consider the interval $\displaystyle [1,2]$ can and the cover $\displaystyle \{ 1-\frac{1}{n},2+\frac{1}{n}\}_{n \in \mathbb{N}}$

This covers [1,2] for all values of n but it also covers for any finite value of n.

Now consider (1,2) and the cover above this covers it, but it must be true for every covering.

Now consider $\displaystyle (1+\frac{1}{n},2-\frac{1}{n})_{n \in \mathbb{N}} $

at "infinity" it covers (1,2) but doesn't for any finite value of n.

so (1,2) is not compact

I hope this helps.

Brett