We cover each of the four possibilities below.
Closed and bounded: $[0,1]$
Closed and not bounded: $\cup_{n\in Z}[2n,2n+1]$
Bounded and not closed: $(0,1)$
Not closed and not bounded: $\cup_{n\in Z}(2n,2n+1)$
Let's talk about real numbers here, rather than general metric or topological spaces. This way we don't need notions of Cauchy sequences or open balls, and can talk in more familiar terms.
We define that a set $X \subset \mathbb{R}$ is open if for every $x \in X$ there exists some interval $(x-\epsilon,x+\epsilon)$ with $\epsilon > 0$ such that this interval is also fully contained in $X$.
An example is the inverval $(0,1) =\{x \in \mathbb{R} : 0 < x < 1\}$. Note that this is an infinite set, because there are infinitely many points in it. If you choose a number $a \in (0,1)$ and let $\epsilon = \min\{a-0, 1-a\}$ then we can guarantee that $(a-\epsilon,a+\epsilon) \subset X$. The set $X$ is open.
A set $X$ is defined to be closed if and only if its complement $\mathbb{R}- X$ is open. For example, $[0,1]$ is closed because $\mathbb{R}-[0,1]= (-\infty,0)\cup(1,\infty)$ is open.
It gets interesting when you realise that sets can be both open and closed, or neither. This is a case where strict adherence to the definition is important. The empty set $\emptyset$ is both open and closed and so is $\mathbb{R}$. Why? The set $[1,2)$ is neither open nor closed. Why?
Best Answer
Intuitively speaking, an open set is a set without a border: every element of the set has, in its neighborhood, other elements of the set. If, starting from a point of the open set, you move away a little, you never exit the set.
A closed set is the complement of an open set (i.e. what stays "outside" from the open set).
Note that some set exists, that are neither open nor closed.