general topology - Is it true that open intervals on the real line and the real line is closed in $mathbb{R}^2$

I am trying to construct a closed connected set in $\mathbb{R}^2$ and I vaguely remember that an open interval is considered closed in $\mathbb{R}^2$ but I lack a satisfactory explanation.

Claim: open intervals on the real line and real line are closed in $\mathbb{R}^2$

The argument was that since each point is surrounded by a ball, and that ball necessarily contains points not in the open intervals, therefore the intervals are not open. But this is not good because it does not imply that they are closed.

Can someone please verify whether open intervals are considered closed in $\mathbb{R}^2$?

Answer

You're right, that argument does not show that they aren't closed, since not being open isn't the same as being closed.

An open interval $(a,b)$ is not closed in $\mathbb{R}^2$ because there are sequences of elements in $(a,b)$ converging in $\mathbb{R}^2$ whose limits aren't in $(a,b)$. Just take a sequence approach $a$ in $(a,b)$. In general this argument shows that things that aren't closed in $\mathbb{R}$ aren't going to all of a sudden become closed in $\mathbb{R}^2$.