PROVING INTERSECTION OF Any number of COMPACT SETS is COMPACT?

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Discussion Overview

The discussion revolves around the question of whether the intersection of any number of compact sets is itself compact. Participants explore this concept within the context of topology, specifically referencing the Heine-Borel theorem and its implications for closed and bounded sets.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant has shown that the intersection of closed sets is closed and is attempting to prove that the intersection of compact sets is bounded, referencing the Heine-Borel theorem.
  • Another participant notes that the intersection of a collection of sets is a subset of each individual set, implying that if the individual sets are bounded, their intersection must also be bounded.
  • A participant raises a question about the context of the discussion, asking whether the sets are in the realm of real numbers or more generally in R^n, and clarifies that compactness in metric spaces requires both closedness and boundedness.
  • One participant challenges the assertion by stating that the Heine-Borel theorem does not hold in all metric spaces, providing an example involving the rational numbers.
  • Another participant suggests that the original poster (OP) is likely working in R^n due to the mention of the Heine-Borel theorem.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the intersection of compact sets remains compact. There is no consensus on the validity of the original poster's argument, and the discussion includes competing perspectives on the definitions and implications of compactness.

Contextual Notes

Some participants highlight the importance of the specific definitions of compactness and boundedness in different topological spaces, indicating that assumptions about these properties may vary based on the context.

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I'm stuck ...
Ive proved the intersection of any number of closed sets is closed ...
and Let S = { A_a : a Element of I } be an collection of compact sets...then
by heine Borel Theorem ...Each A_a in S is closed...so this part is done now I just
have to show the intersection is bounded...
so I said since each A_a in S is bounded by Heine Borel ..then A_a is a subset of [a,b]
and then this is when I am really stuck:
Let B={[a,b] : There exists an A_a in S such that A_a subset of [a,b]}

I took the intersection of both sides of the subset ...and said Intersection S is a subset of
Intersection of [a,b] so Intersection of S is bounded and therefore compact

but somehow I think this is an error ..I don't know how to do this formally

Please help
 
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The intersection of a bunch of sets is a subset of each of the sets. And a subset of a bounded set is <fill blank>.
 
Are you talking about sets or real numbers? Or, more generally, in Rn?

It is true that in any metric topology, a compact set is both closed and bounded.

It is NOT true that in any metric topology, closed and bounded sets are compact.
For example, the Heine-Borel theorem is not true of the rational numbers with d(x,y)= |x-y|.

If you are working in the real numbers, then morphism is giving you a good hint: if A is a bounded set the A intersect ANY other sets is bounded.

If you are working in any topological space, the theorem is still true. You would need to look at the basic definition of "compact" because without a metric, "bounded" is not even defined: let {U} be an open cover for A and show it has a finite subcover.
 
this seems to be false, unless by compact you mean the bourbaki definition of compact which includes hausdorff as part of the definition. if that is what you mean by compact then a compact set is also closed. so since a closed subset of a compact set is compact you get it easily. but the disjoint union of two closed unit discs, then identified except at the origin seems to be a space in which both discs are compact but their intersection is not, being a punctured disc.
 
I think the OP is working in R^n, because he/she mentioned Heine-Borel.
 

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