Compact Sets

**GoldendoodleMom** · October 30th 2008, 10:32 AM

Prove that the union of any two compact subsets of the reals is compact.

**ThePerfectHacker** · October 30th 2008, 10:43 AM

Originally Posted by GoldendoodleMom

Prove that the union of any two compact subsets of the reals is compact.

If $A$ and $B$ are compact sets then there is $R>0$ so that $A,B$ are contained in $\{ x \in \mathbb{R} : |x| < R \}$ . But then $A\cup B$ is contained in $\{ x \in \mathbb{R} : |x| < R\}$ . Therefore, $A\cup B$ is closed. Also the union of two closed sets is closed therefore $A\cup B$ is closed. Thus, $A\cup B$ is compact.

**Moo** · October 30th 2008, 12:30 PM

Hello,

Another method is to use the Heine-Borel theorem (note that $\mathbb{R}$ is a separated space) :

If A is a compact, then there exists a finite family of subsets $(A_i)_{i \in I}$ , where I is a finite set, such that $A \subseteq \bigcup_{i \in I} A_i$
Similarly, $B \subseteq \bigcup_{j \in J} A_j$ , where J is a finite set.

Then $A \cup B \subseteq \bigcup_{k \in I \cup J} A_k$

and $I \cup J$ is finite.

Thus $A \cup B$ is a compact set.

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