jessicaw said:Homework Statement
union of 2 disjoint sets is disconnected
Homework Equations
The Attempt at a Solution
For open disjoint sets it is obvious as it is the definition of disconectedness.
So it remains to prove the closed sets/one open and open closed set case.
In any case, i draw some diagrams and i believe they are disconnected.
A set is considered disconnected if it can be divided into two non-empty subsets, where the intersection of the two subsets is an empty set and the union of the two subsets is equal to the original set. In other words, there is a gap or break in the set that separates the two subsets.
A closed set is a set that contains all of its boundary points, while an open set does not contain any of its boundary points. In other words, every point in a closed set is an interior point, while an open set can have both interior and boundary points.
To prove that a set is disconnected, you can show that it can be divided into two non-empty subsets, where the intersection of the subsets is an empty set and the union of the subsets is equal to the original set. This can be done by finding a point in the set that creates a gap or break, separating the two subsets.
The closed case of disconnectedness refers to a set that is disconnected and also closed. This means that the two subsets that divide the set are both closed sets and the original set is also closed.
Yes, a set can be both open and closed. This is known as an open/closed set. An open/closed set contains all of its interior points and also its boundary points. In other words, every point in the set is both an interior point and a boundary point.