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 $\mathbb{R}^2$. 
 A: To your first question, one definition of a closed set is a set that contains all its limit points. Hence, if a set has no limit points then it is closed.
For your second, $(a,b)$ would be a line segment in the plane - which is not open, since at any point on the line, there are points in the second dimension arbitrarily close to it that are not on the line.
The equivalent notion of open and closed intervals will be open and closed balls - an open ball is $B_\epsilon(x) = \{y \in \mathbb R^2 :  ||y-x||<\epsilon\}$
A: 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,
