0

My question is:

Is $$L=\{(x,y)|y=0\}\cup\{(x,y)|x>0, y=\frac{1}{x}\}\subset\mathbb{R}^2$$ connected or not?

I know: A set is connected iff there no exists $A_1,A_2\subset A$ open sets such that

  1. $A_1\neq\emptyset,A_2\neq\emptyset$,
  2. $A_1\cap A_2 =\emptyset$,
  3. $A_1\cup A_2 = A$.

Well, I propose, $L$ is not connected. That means there exists

$$L_1=\{(x,y)|y=0\}\neq\emptyset \quad\land\quad L_2=\{(x,y)|x>0,y=x^{-1}\}\neq\emptyset,$$ then the intersection of them is emptyset, and the union of $L_1$ and $L_2$ is $L$. So,all points are valid, so $L$ is not connected.

Is my proof right?

MathCracky
  • 833

1 Answers1

1

You have included no demonstration that $L_1$ and $L_2$ are open in the subspace topology on $L$. That is the key point.

The trick here is to find open sets $U_1$ $U_2$ in $\Bbb R^2$ such $L_1 = L \cap U_1$ and $L_2 = L \cap U_2$.

Hint: You might want to think about $f(x) = {1\over 2x}$.

Paul Sinclair
  • 43,643