Let X ={a,b,c,d,e}   and let A={{a,b,d}, {d,c}, {c,e}, X} ba a subbase. Find the base B generated by A. Also, find the topology generated by B.  

`\alpha` = {{a,b,c},{c,d},{d,e}}   be a subbase. Find the base  `\beta` generated by  `\alpha`. Also, find the topology generated by   `\beta`.

Solution:

First Part: 

            Let X = {a, b, c, d, e} and let A = {{a, b, d}, {d, c}, {c, e}, X} be a subbase. To find the base B generated by A, we need to take all possible unions of elements in A.

B = {A ∪ B | A, B ∈ A} 

= {({a, b, d} ∪ {d, c}), ({d, c} ∪ {c, e}), ({c, e} ∪ X), ({a, b, d} ∪ {d, c} ∪ {c, e} ∪ X)} 

= {{a, b, c, d}, {c, d, e}, {a, b, c, d, e}}

            The topology generated by B is the collection of all subsets of X that can be formed by taking arbitrary unions of elements in B. In other words, it is the collection of all subsets of X that can be formed by taking arbitrary unions of the sets in B.

Topology = {S ⊆ X | S = U₁ ∪ U₂ ∪ ... ∪ Uₙ, U₁, U₂, ..., Uₙ ∈ B}

Second Part:

            Let `\alpha` = {{a,b,c},{c,d},{d,e}} be a subbase. To find the base `\beta` generated by `\alpha`, we need to take all possible unions of elements in `\alpha`.

`\beta` = {`\alpha` ∪ `\beta` | `\alpha`, `\beta` ∈ `\alpha`}

= {({a,b,c} ∪ {c,d}), ({c,d} ∪ {d,e}), ({a,b,c} ∪ {c,d} ∪ {d,e})}

= {{a,b,c,d}, {c,d,e}, {a,b,c,d,e}}

            The topology generated by `\beta` is the collection of all subsets of X that can be formed by taking arbitrary unions of elements in `\beta`. In other words, it is the collection of all subsets of X that can be formed by taking arbitrary unions of the sets in `\beta`.

Topology = {S ⊆ X | S = U₁ ∪ U₂ ∪ ... ∪ Uₙ, U₁, U₂, ..., Uₙ ∈ `\beta`}