Example $8.10$ from Rotman shows that $X=\bigvee_{i\geq1}S_i^{1}$ with base point $b$ is not homeomorphic to the subspace $Y$ of $\mathbb{R}^2$ containing the circles $C_n$ have center $(0,1/n)$ and radius $1/n$. We can find a set $F=\{x_n;n\geq1\}$ where $x_n\in C_n\setminus\{0,0\}$ such that $F\cap C_n=\{x_n\}$ for all $n\geq1$. $F$ is closed in $C_n$ and so is closed in $X$. But is not closed in $Y$ because $(0,0)\notin Y$ and it is a limit point of $F$ in $Y$.