Union of a locally finite collection of compact set is compact

Question

I want to know the union of a locally finite collection of compact set is compact.

First I know the following facts(theorems);

0. finite set is compact and finite implies locally finite.

1. Union of a finite collection of a compact set is compact.

2. Every locally finite collection of subsets of a compact set is finite (hence locally finite)

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So the finite union of a locally finite collection of compact set is compact, but how about the general case? I mean countable union of a locally finite collection of compact set is compact?

Answer 1

It is not true. Let $K_n = \{n\} \subset \mathbb R$. The $K_n$ form a locally finite family of compact subsets of $\mathbb R$, but their union is not compact.