No there aren't any others. (If I understand your question correctly.)
How many bijections can you have on $X$? Two, namely $f_1(x) = id_X$ and $f_2: a \mapsto b, b \mapsto a$.
Why is neither of these a homeomorphism between $T_1$ and $T_2$? Because $f^{-1}(\{a\}) = \{a\}$ is not open in $T_1$ hence $f_1$ (and similarly $f_2$) is not continuous in this case.
What about $T_2 \to T_1$? Well, a homeomorphism has to be open but $f_1(\{a\}) = \{a\}$ is not open. So again, neither of the $f_i$ is a homeomorphism.
A point $x \in X$ is said to be near a set $Y \subseteq X$, if every neighborhood of $x$ intersects $Y$.
In the standard (Euclidean metric) topology for $\Bbb R^n$, the near points of an open ball of radius $r< 1$ centered at $a$ are the elements of the open ball, and its boundary (so, a closed ball centered at $a$ of radius $r$). This is loose enough so we get "plenty" of near points, but exclude anything "not too close".
If we take the discrete topology on $\Bbb R^n$ (induced by the discrete metric), an open ball of radius $r < 1$ centered at $a$ contains "just $a$". In other words, in the discrete topology , every other point but $a$ is "far, far away" from $a$.
That is to say, the usual topology and the discrete topology behave qualitatively differently.
As for your question about "size", it turns out that the size of the Euclidean topology on $\Bbb R$ is the same size as $\Bbb R$, whereas the power set is "bigger" (the proof of my first statement rests upon the fact that we can write any open interval as a countable union of open intervals with rational endpoints, using Cauchy sequences, the second statement is due to Cantor, who showed there is no bijection between $X$ and $2^{X}$ for any set $X$).
Best Answer
If $(X,\tau_X)$ and $(Y,\tau_Y)$ are homeomorphic, then there is a bijection between the two spaces. Namely, there is a function $f\colon X\to Y$ which is a bijection, and satisfies that $f[U]$ is open if and only if $U$ is open (for $U\subseteq X$).
Now ask yourself, is there a bijection $f\colon\Bbb R\to\Bbb Z$?