Closed and open sets in metric spaces I'm going to be a freshman and I have just learnt topology recently. Here is my question:
If a set has no limit point, then it is closed?
For example, from which I read in Principles of mathematical analysis of Rudin, a finite set is closed; and the set of integers is also closed. In fact, each of those two sets has no limit point.
Moreover, I want to ask one more question: How is the segment $(a,\,b)$ shown in $\mathbb{R}^2$? Because Rudin also said that a segment $(a,\,b)$ is open in $\mathbb{R}^1$ but it is not in $\mathbb{R}^2$. 

If you mean the interval $]a,b[ \, \times \, \{0\}$ as a subset of $\mathbb R^2$ with the inclusion map $i : \mathbb R \to \mathbb R^2$ given by $i(x) = (x,0)$, then notice that even though $]a,b[$ is open in the real line, the map $i$ is not an open map, i.e. it doesn't map open sets to open sets. The reason why is that if you draw a ball $\{ y \in \mathbb R^2 \, | \, \|x-y\| < r \} \overset{def}= B_r(x)$ around some $x \in i(]a,b[)$, then it will always contain points with non-zero second coordinate, thus cannot be contained in $i(]a,b[)$. 
(I misread your question about closed sets... I'll leave the honor and answer to Mathmo123.)
Hope that helps,
