A set in topology I have a couple of conceptual questions in topology: 
(1) If we have a set $X$ with a topology $T$, do all the subsets of $X$ have to be open?
Here my understanding is that since $X$ has a topology, it is open and so its subsets have to be open.
(2) If $X$ is closed does that mean it can have no topology?
 A: Topologies are collections of subsets of a space $X$ which we define to be open (of course these sets must satisfy the three axioms for a topology). There is no need for every subset of $X$ to be in the topology (which is what we mean by an open set). In fact, spaces in which every subset is open are called discrete spaces, and are not interesting to study.
And a space $X$ is always closed in any topology since the empty set is always open (since it is necessarily a subset of the topology), and the complement $X \setminus X = \varnothing$.
To help you understand better, let's look at one of the simplest non-discrete spaces there is. Let $X = \{a, b\}$, and let the topology $\tau = \{X, \{a\}, \varnothing\}$. The open sets in this space are the sets in the topology, namely $X$ (the space itself), $\{a\}$, and $\varnothing$. Notice that these satisfy the three axioms for a topology:
1) $X$ and $\varnothing$ are open,
2) Any union of open sets is open, and
3) A finite intersection of open sets is open.
The closed sets for this topology are the complements of the open sets. These are $X \setminus X = \varnothing$, $X \setminus \varnothing = X$, and $X \setminus \{a\} = \{b\}$.
In this particular space, every set is either open or closed or both. But this is not necessarily the case. Many spaces have sets which are neither open nor closed. For example, $(0, 1]$ is such a set in $\mathbb R$ with the Euclidean topology.
