Let A=[a1,a2] and B=[b1,b2] and be closed and bounded intervals in R. In each of the following groups determine which of the four possible intersection values can be realized and which cannot for A and B in R. Depict those that can be realized and prove that the reminder cannot.
A – (0,0,0,0),(1,0,0,0),(0,1,0,0),(1,1,0,0)
B – (0,0,1,0),(1,0,1,0),(0,1,1,0),(1,1,1,0)
Let A and B be closed sets in topological space X.
a/ prove that if I_(A,B)=(0,0,0,0) then A∩B=∅
b/ prove that if I_(A,B)=(1,0,0,0) then A∩B=∂A ∩ ∂B
a/ provide an example demonstrating that ∂A need not equal ∂(int(A)).
b/ show that ∂A= ∂(int(A))for regularly closed sets A
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