Is A Finite Union of Perfect Sets Perfect?

[onemachine]

We know that a finite union of closed sets is closed. Thus, a finite union of perfect sets is closed...but is it perfect?

In other words, could a finite union of perfect sets contain isolated points? I cannot think of anyway this can happen...but I also can't prove it...yet.

Any input is appreciated.

 

[pka]

In other words, could a finite union of perfect sets contain isolated points? I cannot think of anyway this can happen...but I also can't prove it...yet.

If \(\displaystyle A~\&~B \) are two perfect sets and \(\displaystyle x\in A\cup B \) then \(\displaystyle x\in A\text{ or }x\in B \).
Can \(\displaystyle x \) be an isolated point?

 

[onemachine]

If \(\displaystyle A~\&~B \) are two perfect sets and \(\displaystyle x\in A\cup B \) then \(\displaystyle x\in A\text{ or }x\in B \).
Can \(\displaystyle x \) be an isolated point?

aha