Open Set definition of a Topological Space, from Jänich's Topology:
From definition (1) for some set $A \subset X$: $$X \setminus A \text{ is open} \implies A \text{ is closed}$$
Further, from Axiom 3, $\emptyset$ and $X$ are both open. So we have that
$$X \setminus \emptyset = X \text{ is open} \\ \text{and} \\ X \setminus X = \emptyset \text{ is open}$$
This implies that $\emptyset$ and $X$ are both open and closed. Is this correct to say?
The answer appears to just simply be yes. I was thrown off by the idea of being both open and closed (this is my first time studying topology), but it seems that this is just natural to the discipline.
Thank you Brian M. Scott!
See Are $\emptyset$ and $X$ closed, open or clopen? for a more thorough discussion.
Yes, in any topological space $(X, \mathcal{O})$, $\emptyset$ and $X$ are open and also closed. (such sets are often called "clopen"). It's unfortunate that this is linguistically paradoxical, and can be a case for confusion for beginning students. A set can be both open and closed (or neither, as is often the case). In the discrete topology (i.e. $\mathcal{O}=\mathscr{P}(X)$) every subset is open and closed.
Some terminology around this: A space $X$ where $\emptyset$ and $X$ are the only clopen sets is called connected. A space where every subset is open or closed (or both) is called a door space. A space where every open subset is the union of clopen subsets is called zero-dimensional. A space where every open subset is also closed (and vice versa) is called a partition space.
