I have been asked to determine if the following statements are ture or false...I have attepmted to answer each question using my understanding of the definitions of compact (a set is compact iff every open cover of S contains a finite subcover), the statement of the Heine-borel Theorem( A subset S of R is compact iff S is closed and bounded), and the Bolzano-Weierstrass theorem( If a bounded sub set S of R contains infinitely many points, then there exists at least one point in R that is an accumulation point of S). Please tell me if I am using reasoning that is not correct for any of them...Thanks(adsbygoogle = window.adsbygoogle || []).push({});

Every finite set is compact.

True. ex [0,3]

No infinite set is compact.

False, but I cannot think of a counter example...

I was thinking that a set could be bounded and still contain infinitley many points, but am not sure if it would be closed.

If a set is compact then it has a max and a min.

False it must also be non empty.

If a set has a max and a min then it is compact.

False, (2,3) has a max and a min but is not ompact.

Some undounded sets are compact.

False due to the hiene-borel theorem

If S is a compact subset of R then there is at least one point in R that is an accumulation point of S

Flase, it need not be compact, just bounded and contain infinitely many points.

If S is compact and x is an accumulation point of S then x is an element of S.

True, Since S is closed it contains all of its accumulation points.

If S is unbounded, then S has at least one accumulation point.

False, S must be bounded, and contain infinitely many points by Bolozano -weierstrass.

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# Homework Help: Analysis (compact or not?)

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