# 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?