The example is following:
Let $Y=\{(0,y):y \in R\}$. Let $E \subset R^2$, i.e., the subset of the real plane, and $E=Y\cup \{ (\frac 1n, \frac k{n^2}): n\in Z^+, k \in Z\}$. The topology on $E$ is this:
- The point $(\frac 1n, \frac k{n^2})$ is open;
- $\{U_n(y_0): n=1,2,...\}$ are the nbhds of the point $(0,y_0)$ of $Y$ is defined as following: $U_n(y_0)=\{(x,y): x \le \frac 1n, |y-y_0|\le x\}$.
My text book said $Y$ is closed discrete in $E$. I could see $Y$ is closed: for any point $y \in Y^C$, the set $\{y\}$ is open which is disjoint with $Y$. However, I fail to show that $Y$ is a discrete space.
Could anybody help me? Thanks ahead:)