What is the difference between saying that $X$ is topological space and $(X,T)$ is topological space? In some theorems I see in our lecture notes, sometimes it mentions that $X$ is a topological space. But isn't $X$ is just a set, and there must be some topology $T$ with $X$ to have a topological space?
 A: It is exactly the same, just that you write the couple $(X, \tau)$ when you want or need to emphasize there is a topology equipped to the set $X$.
When the context is clear, it is a standard abuse of notation to write only the set $X$ saying that it is a topological space, without writing the topology. But you are right: rigorously one should always write $(X, \tau)$.
Note that there are even some topological spaces that are even equipped by a distance and the topology you endow the set with is not the one induced by the distance: you understand that in this example it is somehow important to specify that you are considering the space $(X, \tau, d)$. There are even spaces called “metric measure spaces”: topological spaces where the topology is induced by the distance and equipped with a measure function defined on a sigma algebra $\mathcal{M}$, with some compatibility properties between them: there one usually writes $(X, \mathcal{M}, d)$. So, when the context is clear and “we know where we live”, it is common to refer saying just $X$, keeping in mind the structure we give to the set $X$ when defining it.
A: It's an example of synecdoche: using a part to represent the whole. It is clear that if we refer to $X$ as a topological space, there is an underlying topology $T$ associated to it, without having to spell it out.
For instance, technically a group is a tuple $(G, \cdot, e, ()^{-1})$, as it is a set with an operation with an identity and an inverse, but for simplicity we abuse notation and simply call it $G$. This is standard in mathematics.
A: It is standard to say that $(X,\tau)$ is a topological space, and it is customary to say that $X$ is a topological space when the associated topology $\tau$ is understood by the context, or when specifying $\tau$ is not that important in the statements.
See my previous answer to make things clear.
