Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected. (Hint:Show that given $x_0 \in U$, the set of points can be joined to $x_0$ by a path in $U$ is both open and closed in $U$.)
as i found the answer here Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected but i did n't the red line
Pliz help me,,,,,,,,
When you want to show that $A$ is closed, one way to do so is to show that the complement is open. So, you take some $x\in A^c = U\setminus A$ (the complement in U) and show that there exists some $\varepsilon>0$ small enough so that $B(x,\varepsilon)\subseteq U\setminus A$ (the ball centered at $x$ with radius $\varepsilon$ is contained in $U\setminus A$).