QuestionQuestion

3/ Determine Int(A) and CL (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^2xϵR} in R^2 with the standard topology.

4/ Consider the particular point topology 〖PPX〗p on a set X. Determine int(A) and CL(A) for sets A containing p and for sets A not containing p.

6/ For sets A and B in a topological space X prove that:
CL(X-A)= X-Int (A) and Int(A)∩Int(B)=Int(A∩B).

7/ In each case, determine whether the relation in the blank is "⊏ ,⊐" or = . In each cases where equality does not hold, provide an example indicateing so.
CL(A)∩CL(B)………………CL(A∩B).
CL(A)∪CL(B)………………..CL(A∪B).

8/ For each n∈z_+ let B_(n={n,n+1,n+2,……} ) and consider the collection B={B_n |n ∈z_+ }.
Show that :
B is a basis for a topology on z+.
The topology on X is generated by B is not haousdorff.
The sequence{2,4,6,8,……} converges to every point in Z+ with the topology generated by B.
Prove that every injective sequence converges to every point in Z+. with the topology generated by B.

9/ Determine the set of limit points of [0,1] in the finite complement topology on R.

11/ Determine the set of limit points of A={1/m+1/n∈R|m,n∈z_+ } in the standard topology on R.

12/ Show If (xn) is an injective sequence in R, then(xn ) converges to every point in R with the finite complement topology on R.

13/ Let A be as subset of Rⁿ in the standard topology, If x is a limit point of, then there is a sequence of points in A that converges to x.

14/ Let τ be collection subsets of R consisting of the empty set and every set whose complement is countable.

Show τ is topology on R. (it is called countable complement topology).
Show that point 0 is a limit point of the set A= R-{0} in the countable complement topology.
Show that in A= R-{0} there is no evidence converging to 0 in the countable complement topology.

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