I believe the following definition is correct for a topological space. However, the book I am using refers to just $X$ being a topological space later on down the road which I am confused with.
$\textbf{Question about Terminology:}$ Is the topological space $(X, \tau)$ really the set $X$? I would have thought it would have been $\tau$. I assume the two tuple here acts similar to that of a groupoid where it is really just a set but I could be wrong.
$\textbf{Definition:}$ Let $X$ be a non-empty set. Also, let $\tau=\lbrace O_i\subseteq X| i\in I\rbrace$ ($O$ signifies these sets are open, i.e. not necessarily open in the context of metric spaces).
We say $(X, \tau)$ is a $\textbf{topological space}$ iff the following conditions are satisfied:
$(C1)$ $\phi, X\in \tau$
$(C2)$ $\forall J \subseteq I$, [if $J$ is finite, then $\cap_{j\in J} O_j\in \tau$]
$(C3)$ $\forall K \subseteq I$, $\cup_{k\in K} O_k \in \tau$.
Formally, the topological space is the pair $(X, \tau)$ with $\tau$ the collection of open sets that obey the axioms. But the $X$ is important too, as it provides the context: all open sets are subsets of $X$ (and we know that $X \in \tau$ too).
$\tau$ is called a "topology on $X$", and the pair together a "topological space".
It is common practice to omit the $\tau$ and consider it understood in "the topological space $X$". The context must make clear what topology $\tau$ is actually meant. In measure theory the same is often done and the $\sigma$-algebra is also understood from context (the pair of set $X$ and $\sigma$-algebra is then a "measurable space"). Likewise with metric space (this includes the $d$, so is also a pair $(X,d)$) and a uniform space.
In algebra you also say the "ring $\mathbb{Z}$" instead of the more formal "ring $(\mathbb{Z}, +, \times, 0, 1)$" etc. Only when we want to be extra precise or when we have different structures on the same set, the extra structure is emphasised this way, usually.