# proving a set is closed

• Dec 14th 2010, 12:59 PM
cottekr
proving a set is closed
how can I prove that [a, b] is a closed set?
• Dec 14th 2010, 01:07 PM
snowtea
A set is closed if its complement is open.

In this case prove:
(-inf, a) union (b, +inf)
is open.

To prove this set is open just use the definition.
Intuitively: show for every point in the open set, there is a small neighborhood around that point that is also in the set.

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You can also use the lemma that a set is closed iff the closure is the set itself. I.e. show the closure of [a,b] is just [a,b].
• Dec 14th 2010, 01:14 PM
Drexel28
Quote:

Originally Posted by cottekr
how can I prove that [a, b] is a closed set?

It depends what your definition of closed is!
• Dec 23rd 2010, 03:38 PM
alexandros
Quote:

Originally Posted by cottekr
how can I prove that [a, b] is a closed set?

To prove that [a,b] is closed you must prove:

1) For all x>b there exists an ε>0 ,such that:

......For all ,y : $|x-y|<\epsilon\Longrightarrow y>b$.

2) For all x<a there exists an ε>0 , such that:

......For all ,y : $|x-y|<\epsilon\Longrightarrow y