Example: intersection of compact sets which is NOT compact

  • #1

Homework Statement

Let S = {(a,b) : 0 < a < b < 1 } Union {R} be a base for a topology. Find subsets M_1 and M_2 which are compact in this topology but whose intersection is not compact.

Homework Equations

The Attempt at a Solution

I'm not even sure what it means for an element of S to be compact, so I haven't been able to make any attempt at a solution.

Answers and Replies

  • #2
I assume you're working on the real line, right?

Do you know the meaning of the terms 'open set', 'cover' and ' finite subcover'?
I don't mean to be condescending; just to know.
  • #3
I assume it is the real line, and so the topological space will be (R,S).

Yes I do know what open set/cover/finite sub cover mean
  • #4
The most general definition is that a subset S is compact iff (def.) every cover of S by open sets has a finite subcover. There are more specialized results, e.g., for R^n, compactness is equivalent to being closed and bounded,and, for metric spaces you have, e.g., every sequence has a convergent subsequence, but the first one covers all cases.

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