Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How many components of complement of a compact and connected set in $\mathbb{R^2}$has?

Are there any tricks to solve the problem?

share|improve this question
add comment

2 Answers 2

up vote 2 down vote accepted

The complement of a compact, connected set in $\Bbb R^2$ can have any finite number of components, and it can also have countably infinitely many components. The complement of a line segment has one component. The complement of a circle has two. The complement of a figure $8$ has three. By adjoining more and more circles, you can get any finite number. Finally, if you take the union of circles of radius $\frac1n$ and centre $\left\langle\frac1n,0\right\rangle$, you get a compact, connected set whose complement has infinitely components.

share|improve this answer
add comment

It could be anything. Consider a closed disc with $n$ separated open discs deleted from it. Such a set is closed and bounded, hence compact; it is clearly connected. But its complement has $n+1$ connected components.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.