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Given are the following sets

$A=$ { $(x,y)| x>0, y<0$ }

$B=$ { $(x,y)| x<0, y>0$ }

$C=$ { $(x,y)| x^2+y^2=0$ }

I need to tell that the union of the sets A , B and C is connected or not, but im not sure if I chose the right way to prove that is connected. I think that its enough to show that the union of these sets is path connected and so it implies that the set is connected.Can anyone tell me if I'm right this way?Or is there another way of solving this problem?

Thank you for helping!

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    Are you sure your definition of $C$ is correct? As it is now, $C=\{(0,0)\}$ (assuming these are subsets of $\mathbb{R}^2$). – 2017-02-03
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    Also, the answer to the question depends on which topology you equip your space with. I think you should point out that these are subsets of $\mathbb{R}^2$, and that you are using the standard topology (assuming these things are true) – 2017-02-03
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    Yes I'm sure the problem is given like this, graphically we have the union of the 2-nd and 4-th quadrant and the point $(0,0)$ on the real plane so every point of this set is path connected through $(0,0)$ , and yes it has the standard topology. – 2017-02-03

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I assume these sets are subsets of $\mathbb{R}^2$ with the standard topology, hence $C=\{(0,0)\}$. It is clear that both $A$ and $B$ are path connected (they are the 2nd and 4th quadrant of $\mathbb{R}^2$ respectively). With these premises, you can clearly show that $A\cup B\cup C$ is path connected. Indeed, any two points $a\in A$, $b\in B$ can be connected by the path

$$ \gamma:I\rightarrow\mathbb{R}^2::t\mapsto \begin{cases} (1-2t)a & 0\leq t\leq1/2\\ (2t-1)b & 1/2

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    Clear :D . Thank you very much! :) – 2017-02-03
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    You're welcome :) – 2017-02-03