The role of the topology in a topological space. Consider the topological space $(X, T)$ where $X$ is a set and $T$ is a topology which is a family of open subsets of $X$. I learned that the continuity of a function is determined by the topology. Say, the continuity of a function from $(X,T_{1})$ to $(X,T_{2})$ is determined by the relation between $T_{1}$ and $T_{2}$ even though its domain and codomain remain the same. $T_{1}$ and $T_{2}$ are families of the open subsets of $X$ by definition.
My question: Some open sets in $T_{1}$ need not be contained in $T_{2}$ because a set can have different topologies. In that case, are such sets not considered as open sets in $(X,T_{2})$?
 A: There is some confusion when you say that “$T$ is a topology which is a family of open subsets of $X$”. This is true, but the whole truth is that $T$ is the family of all open subsets of $X$. So, if $T_1\not\subset T_2$, and if $O\in T_1\setminus T_2$, then $O$ is not an open subset of $(X,T_2)$.
A: The answer to your question is yes, such sets need not be "considered as open in $T_2$. To put it more precisely, there are elements of $T_1$ which are not elements of $T_2$ (recall that elements of $T_1$ and $T_2$ are themselves subsets of $X$).
Consider $(\mathbb{R},T_1)$, $(\mathbb{R},T_2)$ and $(\mathbb{R},T_3)$ where $T_1$ is the usual Euclidean topology, $T_2$ is the discrete topology, (so every subset $S$ of $\mathbb{R}$ is in $T_2$), and $T_3$ is the coarse topology (the only elements of $T_3$ are $\mathbb{R}$ and $\emptyset$).
Now, for example, any function $f:(\mathbb{R},T_2) \rightarrow (\mathbb{R},T_1)$ is continuous. Here there are many elements of $T_2$ which are not contained in $T_1$.
The point here is that the topology is an entirely separate thing from the underlying set. I start with the set $\mathbb{R}$, and this does not come with a topology. It is just a set, in the same way that $\{a,b,c\}$ is just a set (despite the fact that $\mathbb{R}$ is a lot bigger). I must equip it with a topology. Once I've done that, I have a topological space. I can create a different topological space, with the same underlying set, by equipping $\mathbb{R}$ with a different topology (and that's what I've done above). Now that I have these two topological spaces, I can start talking about continuous maps from one of them to the other.
