00:01
Hello students today we will discuss about this question in this question we are we need to prove that the intersection of two open sets is compact if and only if it is empty so here and here we need to find if the intersection of a infinite collection of open sets be non empty compact set or not.
00:40
So first of all, suppose u1 and u2 be to open subset, open subset such that its intersection u1 and u2 is compact since every compact subset is closed and bounded.
01:05
So u1 and u2 is also closed and bounded.
01:17
So now we will prove the contradiction by the contradiction that u1 and u2 is in t subset.
01:26
So if u1 and u2 is not equals to 5 and x belongs to u1 and u2 be arbitrary.
01:39
So that implies that x belongs to u1 and x belongs to u2 since u1 is open and x belongs to u1, so that u1, so that is open and x belongs to u1, so so there exists, dale 1 is greater than 0 such that x minus dale 1, x plus del 1 is a subset or equals to u1.
02:01
And it is same for x belongs to u2.
02:05
So here, del 2 is greater than 0 such that x my del 2, x plus del 2 is a subset or equals to u2.
02:16
So therefore if we choose, dale that is a subset or equals to u2...