Prove or give counterexample, is $K$ a compact set.

Question

Prove or give counterexample, is $K$ a compact set?

$K\colon=\{ n+\frac{1}{n+1}:n\in \mathbb{N}\}\subseteq\mathbb{R}$ in natural topology.

Answer 1

I found the concept of compactness difficult to hold onto for a long time.

The definition: A set is compact if every open cover has a finite sub-cover.

We can show that the set is not compact if we can find an open cover that does not have a finite sub-cover.

Suppose our open cover is a set of intervals $(n,n+\frac {2}{n+1})$

Each of our intervals cover only one point in $K$ and there are infinitely many points in $K.$ There is no finite subset of intervals that covers all of $K.$

Something to think about, every compact set is closed and bounded. It is worth trying to prove each of these. K is not bounded.

The set $K = \{ \frac 1n| n \in \mathbb N\}$ is bounded, but it is not closed.

The set $K = \{0\}\cup \{ \frac 1n| n \in \mathbb N\}$ is closed and bounded and compact.