Chapter 7: Connected and Compact

Question 1

Connected

Answer 1

A nonempty metric space (X, d) is connected if the only subsets that are both open and closed are the empty set and X itself

Question 2

Facts about connectedness

Answer 2

1. Let (X,d) be a metric space. A nonempty set S in X is not connected if and only if there exists open sets U1 and U2 in S such that the intersection of the three is equal to zero, but the intersect of each of U1 and U2 with S is not zero and S is the union of the intersect of U1 and S and U2 and S.

Question 3

More facts about connectedness

Answer 3

2. a set S in Reals is connected if and only if it is an interval or a single point

Question 4

Closure

Answer 4

Let (X,d) be a metric space and A in X. then the closure of A is the intersection of all closed sets that contain A.

Question 5

Facts about Closures

Answer 5

1. the closure of A is closed
2. If A is closed then A = its closure
3. Let (X, d) be a metric space and A in X. Then x in the closure if and only if for every delta >0, B(x,delta) intersect A is not equal to the empty set

Question 6

Interior

Answer 6

Let (X,d) be a metric space and A in X: the interior of A is the set A0 = {x in A: there exists a delta>0 such that B(x,delta) in A}

Question 7

Boundary

Answer 7

partial A := Closure of A \ Interior of A