Let $X=\{1,2,3\}$. Find all topology of $X$ which are either $T_0$, or $T_1$, or $T_3$ Let $X=\{1,2,3\}$. Find all topology of $X$ which are either $T_0$, or $T_1$, or $T_3$
Here is what I got
$T_1$ is discrete topology
$T_0$ are all 1-one point topologies  including
$\{1\},\{2\},\{3\},
\{\{1\},\{1,2\}\},\{\{1\},\{1,3\}\},\{\{1\},\{2,3\}\},\{\{2\},\{1,2\}\},\{\{2\},\{1,3\}\},\{\{2\},\{2,3\}\},\{\{3\},\{1,2\}\},\{\{3\},\{1,3\}\},\{{3\},\{2,3\}\}, 
\{\{1\},\{1,2\},\{1,3\}\},\{\{2\},\{1,2\},\{2,3\}\},\{\{3\},\{1,3\},\{2,3}\}\}$
$T_3$ I'm not so sure.
I also have a question. I know that if $X$ is $T_1$ then $X$ is $T_0$, but is it true that if $X$ is $T_2$ then $X$ is $T_1$?
 A: Generally, $T_i$ implies $T_j$ for $j < i$. (Although some care is needed, because some people would not necessarily understand a $T_i$ space for $i > 0$ to be $T_0$, too)
So, there are two settings here: $T_0$ spaces with 3 elements, and $T_1$ spaces with 3 elements. Any finite $T_1$ space is already discrete, as open sets are closed under finite intersections. Thus, there is only one $T_1$ topology on $\{1,2,3\}$.
For $T_0$, we can try to construct a minimal $T_0$ topology up to permutation of the elements. It is ok if one element has only $X$ as neighborhood, wlog let that one be 3. Both 1 and 2 need smaller neighborhoods, in particular this implies that $\{1, 2\}$ is open. Then again, for one point this can be the smallest neigherborhood, wlog let that be 2. Then, $\{1\}$ is open, too.
So, the $T_0$ topologies are those that (after some permutation) include the open sets $\{1\}$ and $\{1,2\}$.
A: \begin{align*}
\\T1=\{\{1\},¤, X\}
\\T2=\{\{2\},¤,X\}
\\T3=\{\{3\},¤, X\}
\\T4=\{\{1,2\},¤, X\}
\\T5=\{\{2,3\},¤, X\}
\\T6=\{\{1,3\},¤, X\}
\\T7=\{\{1\}, \{1,2\},¤, X\}
\\T8=\{\{2\}, \{1,2\},¤, X\}
\\T8=\{\{1\},\{1,3\},¤, X\}
\\T10=\{\{3\},\{1,3\},¤, X\}
\\T11=\{\{2\}, \{2,3\},¤, X\}
\\T12=\{\{3\}, \{2,3\},¤, X\}
\\T13=\{\{1\}, \{2,3\},¤, X\}
\\T14=\{\{2\}, \{1,3\},¤, X\}
\\T15=\{\{3\},\{1,2\},¤,X\}
\\T16=\{\{1\}, \{2\}, \{1,2\},¤, X\}
\\T17=\{\{2\}, \{3\}, \{2,3\},¤, X\}
\\T18=\{\{1\},\{3\}, \{1,3\},¤, X\}
\\T19=\{\{1,2\}, \{1,3\}, \{1\},¤, X\}
\\T20=\{\{1,2\}, \{2,3\},\{2\},¤, X\}
\\T21=\{\{1,3\}, \{2,3\}, \{3\},¤, X\}
\\T22=\{\{1\}, \{2\}, \{1,2\}, \{2,3\},¤, X\}
\\T23=\{\{1\}, \{3\}, \{1,3\}, \{3,2\},¤, X\}
\\T24=\{\{2\}, \{3\}, \{2,3\}, \{3,1\},¤, X\}
\\T25=\{\{1\}, \{2\}, \{1,2\}, \{1,2,3\},¤, X\}
\\T26=\{\{2\}, \{3\}, \{2,3\}, \{1,2,3\},¤, X\}
\\T27=\{\{1\}, \{3\}, \{1,3\}, \{1,2,3\},¤, X\}
\\T28=\{¤, X\}
\\T29=\{\{1,2,3\},¤, X\}
\end{align*}
So there are 29 topologies of X={1,2,3}
