Empty Topology? I am familiar with the topology on the empty set , Is there a space let's say with the set $R$ where non of its subsets are open ? If it does it will apply that all of them are close no ? and that countable union of them will be close too , which means that this is not a topological space ?
 A: For a set $X$ with topology $\tau$, these must be fulfilled (fulfilling these is what it means for $\tau$ to be a topology)


*

*Any element of $\tau$ is a subset of $X$.

*$X\in \tau$ and $\emptyset \in \tau$.

*For any $U, V \in \tau$, we must have $U\cap V \in \tau$.

*For any $\mathscr U \subseteq \tau$, we must have $\left(\bigcup_{U \in \mathscr U}U\right) \in \tau$.


We see that $\tau = \{X, \emptyset\}$ is the smallest possible topology on any set $X$, since $2$ has to be fulfilled. It's usually called the trivial topology.
Also, as for the connection between open and closed sets, it is not the case that non-open sets are closed. It's rather that the complement of any open set is closed (and vice versa). So the whole set and the empty set are by definition also closed. If these are the two only subsets of $X$ to be both open and closed ("clopen"), then we say that $X$ is connected. So there is no contradiction or disjunction between openness and closedness. They are two different things that may or may not be fulfilled for any given subset. In fact, sets being both at the same time are an important part of general topology.
