Topology Problems

Q1/Determine ∂A in each case
A=(0,1]in the lower limit topology on R
A={a} in X={a,b,c} with topology {X,∅,{a},{a,b}}
.A={a,c} in X={a,b,c} with topology {X,∅,{a},{a,b}}
A={b} in X={a,b,c} with topology {X,∅,{a},{a,b}}
A=(-1,1)∪{2} in the standard topology on R
A=(-1,1)∪{2} in the lower limit topology on R
A={(x,0)ϵR^2 |xϵR}with the standard topology
A={(x,0)ϵR^2 |xϵR} with the vertical interval topology

Q2/for nϵZ determine ∂({n}) in the digital line topology considering separately the cases where n is even and n is odd. Discuss how your results for ∂({n}) reflect the digital image display structure modeled by the digital line?

Q3/Determine the boundary of each the following subsets of R² in the standard topology
A={(x,x)ϵR^2 |ϵR}
A={(x,y)ϵR^2 |x>0,y≠0}
A={(1/n,0)ϵR^2 |n ϵz_+ }
A={(x,y)ϵR^2 |0≤x^2-y^2<1}

Q4/Determine ∂([0,1]) in R with the finite complement topology. Justify your result.?

Q5/
a) Give an example to show that the implication A=C-B→A∪B=C fails in general.
b) Show that if B⊆C then the implication in (a) holds.

Q6/
a) Show that in general the implication A∪B=C→A=c-B fails.
b) Under what condition on A and B in implication in ( a ) is true?

Q7/ Show that ∂A ∪ in A=cl A implies in A=clA-∂A

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