1. Show that B={[a,b)⊂R a&lt;b is a basis for a topology on...

Question

1. Show that B={[a,b)⊂R a<b is a basis for a topology on R.

2. Determine which of the the following collections of subsets of are bases:
a) C1={(n,n+2)⊂R ,n ∈Z
b) C2={[a,b]⊂R,a<b
c) C3={[a,b]⊂R,a≤b
d) C4{(-x,x)⊂R,x∈R
e) C5={(a,b)∪{b+1}⊂R,a<b

3. Determine which of the following are open sets in R_(l ) in each case prove your assertation.
A=[4,5)
B={3}
C=[1,2]
D=(7,8)

4. In arithmetic progression in Z is a set A(a,b)={……,a-2b,a-b,a,a,a+b, a+2b,….} with a,b ∈ Z and b≠0 prove that the collection of arithmetic progression topology on Z.

5. Prove that, in a topological space X if U is open and C is closed then U-C is open and C – U is closed.

6. Which sets are closed sets in the finite complement topology on a topological space X?

7. Which sets are closed sets in excluded point topology EPXp on a set X?

8. Which sets are closed sets in particular point topology PPxp on a set X?

9. Show that a single-point set{n} is closed digital line topology if and only if n is even?

10. Prove that intervals of form [a,b) are closed in the lower limit topology on R?

11. Prove that on a finite set, the discrete topology is the only topology that is Housdorff?

12. Show that R in the finite complement topology is not Housdorff?

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