Question about the definition of topology On James. E. Munkres page 76 in the definition of topology it's written that we say that a subset U of X is open if U belongs to the toplogy on the set X.  Does this mean if we know that a subset of $X$ is open we can't say that the subset belongs to the topology of $X$?? Any help would be appriciated.  Thanks.
 A: It is to say that rather than a topology being the set of open sets of $X$,
the sets of the topology are defined to be open.
For example, $X=\{1,2,3\}$ and let's say its topology is $\tau=\{\emptyset,X,\{1,2\}\}$.
This defines a topology. But what can we say from this? That $\{1,2\}$ is open because $\{1,2\}\in \tau$ but say $\{2\}$ is not open since $\{2\}\notin \tau$
A: No, the statement

We say that a subset $U$ of $X$ is open if $U$ belongs to the toplogy on the set $X$.

is the definition of an open set. So that $U$ is an open set is the same as to say that $U$ belong to the topology on the set $X$.
A: A set is open based on the topology you are using. For instance, in the standard euclidean topology, the interval $(a,b)$ is open. However, if we give $\mathbb{R}$ the indiscrete topology then the only open sets are $\mathbb{R}, \emptyset$.  Hence, if you know a set to be open, it is due to the fact that you understand the topology and deduced the set to be open by the description of the topology. It will be annoying, but soon you'll find yourself beginning every question first by asking, what is the topology?
