This is a tutorial over some topics from the topology of real numbers:

1. A perfect set is closed and contains no isolated points

Notice how the empty set is perfect, but singletons aren't.

2. Nonempty Perfect sets are uncountable.

3. Iff A and B are separated sets, the intersection of the closure of A and B is empty. Likewise, the intersection of A and the closure of B is nonempty.

4. Iff E is disconnected, then there exists separated sets, A B, such that A union B is E

5. E is connected if E is not disconnected

6. E is connected iff for all nonempty disjoint sets, A,B, such that A union B = E, then there exists a sequence in A that converges to an element in B, or a sequence in B that converges to an element in A.

7. E is connected iff for 2 distinct elements in E, there exists a 'c' such that a

Question

This is a tutorial over some topics from the topology of real numbers:

1. A perfect set is closed and contains no isolated points

Notice how the empty set is perfect, but singletons aren't.

2. Nonempty Perfect sets are uncountable.

3. Iff A and B are separated sets, the intersection of the closure of A and B is empty. Likewise, the intersection of A and the closure of B is nonempty.

4. Iff E is disconnected, then there exists separated sets, A B, such that A union B is E

5. E is connected if E is not disconnected

6. E is connected iff for all nonempty disjoint sets, A,B, such that A union B = E, then there exists a sequence in A that converges to an element in B, or a sequence in B that converges to an element in A.

7. E is connected iff for 2 distinct elements in E, there exists a 'c' such that a<c<b and c is in E.

8. F-set is the countable union of closed sets
G-set is the countable intersection of open sets

9. Intersection of dense, open sets is nonempty

10. G is dense iff for 2 arbitrary real numbers, a,b, there exists x in G such that a<x<b

11. G is dense iff the closure of G is the set of real numbers
E is nowhere dense iff the closure of E does not contain open intervals

12. The set of real numbers can't be written as countable union of nowhere dense sets.

@Max1oof

mhchen · Answer

this is math btw