Find the closure and derived set of $ A $ in some topologies.

Let $ X =[0,1] subset mathbb {R} $, let $ mathscr {B} $ to be a base for the topology $ mathscr {T} _ {1} $ sure $ X $ given by $$ mathscr {B} = { {0 }, {1 } } cup {(0,2 ^ {- k}) ,: , k in mathbb {Z} _ { ge0} }, $$and let $ mathscr {T} _ {2} $ to be a topology on $ X $ given by $$ mathscr {T} _ {2} = left {G subset X ,: , frac {1} {2} not in G right } cup {G subset X ,: , (0,1) subset G }. $$Yes $ A = left { left ( frac {1} {4}, frac {1} {2} right) right $ is a subset of the product space $ X ^ { ast} = (X, mathscr {T} _ {1}) times (X, mathscr {T} _ {2}) $, find the closure $ overline {A} $and the derived set $ A $ in $ X ^ { ast} $.

Find $ A $, I've watched the $ A = left { frac {1} {4} right } times { frac {1} {2} $ and using the formula for $ A $ in the product space like $ A = left ( left { frac {1} {4} right } times { frac {1} {2} } right) & # 39; = left (, overline { left { frac {1} {4} right }} times { frac {1} {2} } & # 39; right) cup left ( left { frac {1} {4} right } times overline { { frac {1} {2} }} , right) = left ( left[Frac{1}{4}1right)times{frac{1}{2}}right)cupleft(left(frac{1}{4}1right)times{frac{1}{2}}right)=left[Frac{1}{4}1right)times{frac{1}{2}}$[Frac{1}{4}1right)times{frac{1}{2}}right)cupleft(left(frac{1}{4}1right)times{frac{1}{2}}right)=left[Frac{1}{4}1right)times{frac{1}{2}}$[frac{1}{4}1droite)times{frac{1}{2}}droite)cupleft(left(frac{1}{4}1droite)times{frac{1}{2}}right)=left[frac{1}{4}1droite)times{frac{1}{2}}$[frac{1}{4}1right)times{frac{1}{2}}right)cupleft(left(frac{1}{4}1right)times{frac{1}{2}}right)=left[frac{1}{4}1right)times{frac{1}{2}}$

But the solution is $ A = left ( frac {1} {4}, 1 right) times { frac {1} {2} $.

Where am I wrong? or can not use this formula for a subset of a point in the product space?