# Interior, closure, open set in topological space

• Mar 1st 2011, 12:22 PM
Interior, closure, open set in topological space
$\displaystyle T=\mathbb R^2 \: A=\mathbb Q^+ \times \mathbb R^+$

$\displaystyle \overline{A}=?$ /closure of A/
$\displaystyle Int A=?$
In $\displaystyle (A, \tau_A)$ is $\displaystyle A$ open? $\displaystyle (A, \tau_A)$ /topological space/

• Mar 1st 2011, 12:57 PM
girdav
You can show that if you have two topological spaces $\displaystyle (X,\mathcal T_1)$ and $\displaystyle (Y,\mathcal T_2)$ and if you consider the product space $\displaystyle (X\times Y,\mathcal T\otimes \mathcal U)$ you will have for any $\displaystyle A\subset X$ and $\displaystyle B\subset Y$ that $\displaystyle \overline{A\times B}=\overline{A}\times \overline B$ and $\displaystyle \mathrm{Int}(A\times B)=\mathrm{Int}(A)\times \mathrm{Int}(B)$.
• Mar 2nd 2011, 12:44 PM
Quote:

Originally Posted by girdav
You can show that if you have two topological spaces $\displaystyle (X,\mathcal T_1)$ and $\displaystyle (Y,\mathcal T_2)$ and if you consider the product space $\displaystyle (X\times Y,\mathcal T\otimes \mathcal U)$ you will have for any $\displaystyle A\subset X$ and $\displaystyle B\subset Y$ that $\displaystyle \overline{A\times B}=\overline{A}\times \overline B$ and $\displaystyle \mathrm{Int}(A\times B)=\mathrm{Int}(A)\times \mathrm{Int}(B)$.

He's saying that if you consider giving $\displaystyle \mathbb{R}^2$ the usual topology induced by the usual norm then the induced topology is the same as if $\displaystyle \mathbb{R}^2$ the product topology (when both copies of the reals are endowed with the usual topology). That said, for the product topology it's easy to prove things like $\displaystyle \text{cl}_{\mathbb{R}\times\mathbb{R}}\left(A\time s B\right)=\text{cl}_{\mathbb{R}}(A)\times\text{cl}_ {\mathbb{R}}(B)$. So now since $\displaystyle \mathbb{Q},\mathbb{R}$ are both dense in $\displaystyle \mathbb{R}$ you can conclude that $\displaystyle \mathbb{Q}\times\mathbb{R}$ is dense in $\displaystyle \mathbb{R}^2$. etc.