Let the usual topology $U$ over $\mathbb R$
Say $(\mathbb R^2, T) = (\mathbb R, U) \times (\mathbb R, U)$
$Y = \mathbb R^2 -A^2$ for irrational number set, $A$.
Find the component of the $O(0,0)$ in $(Y,T_y)$
(Here the $T_y$ is a relative topology)
Someone who gave me this question said The answer is $Y$.
My thought)
But I don't know why does it the component. Should be answer $ Y $is unconnected and also component is $(0,0)$ itself?
No, it should not: $Y$ is in fact path-connected. Let $(x,y)\in Y$. This means that either $x\in\Bbb Q$ or $y\in\Bbb Q$. In the first instance, the function $$f(t)=\begin{cases}(2tx,0)&\text{if }0\le t\le\frac12\\ (x,(2t-1)y)&\text{if }\frac12<t\le 1\end{cases}$$
is an arc in $Y$ from $0$ to $(x,y)$. In the second instance, the function $$g(t)=\begin{cases}(0,2ty)&\text{if }0\le t\le\frac12\\ ((2t-1)x,y)&\text{if }\frac12<t\le 1\end{cases}$$ is.