Applying T0 Space Rules

Question

I am starter at Topology. I have found a definition of $T_0$ as; "A topological space $X$ is said to be a $T_0$ space if for any $x,y \in X$, $x \neq y$ there exist an open set $U$ such that $x \in U$ but $y \notin U$"

For $X=\{a, b, c\}$ $\mathcal{T}=\{\emptyset, \{b\}, \{c\}, \{a,b\}, \{b,c\}, \{a,b,c\}\}$ is known $T_0$. However, we cannot find any set $S$ such that ; $a \in S$ and $b \notin S$. So, how can above $\mathcal{T}$ be $T_0$? Thanks.

Answer 1

For $a$ and $b$ we take $S = \{b\}$, then $b \in S$ and $a \notin S$. We cannot do it the other way round (any open set that contains $a$ also contains $b$) but for $T_0$ we only need to be able to do one of them, otherwise the space is called $T_1$.

Answer 2

Ohh I understand now. So the thing that I tried to find was T1 and we can say it's not T1. Thank you so much for answers