Question

Let A be a subset of the space X. Then Bd(A)= Aˉ ∩ X\A. Proof: Take x∈ Aˉ∩ X\A and show every open set in X which contains x also contains a point of A and of X\A. This may be done indirectly. A similar procedure shows the other subset inclusion

Answer 1

The question deals with proving that the boundary of a set A in the context of topology is the intersection of the closure of A and the closure of its complement.

Step-by-step explanation:

The question pertains to the concept of boundary of a set in topology, specifically concerning the proof that the boundary of a set A is the intersection of the closure of A and the closure of the complement of A within a space X. The boundary of a set A, denoted by Bd(A), consists of points that can be approached both by points in A and points not in A (X\A). To prove that Bd(A) = Aˉ ∩ X\A, one must show that any open set around a point x in Bd(A) contains points from both A and X\A. The student is also presented with examples from different mathematical contexts, such as set theory with the union and intersection of sets, and vector algebra with the vector product and volumes described by vectors.