$(X,\tau)$ will be a $T_0$-space but not a $T_1$-space

Question

Exercise: A topological space is said to be $T_0$-space if for each pair of distinct points $a,b$ in $X$, either there exists an open set containing $a$ and not $b$, or there exists an open set containing $b$ and not $a$.A topological space $(X,\tau)$ is said to be $T_1$-space if every singleton set $\{x\}$ is closed in $(X,\tau)$

Put a topology on the set $X=\{0,1\}$ so that $(X,\tau)$ will be a $T_0$-space but not a $T_1$-space.

If I consider the topology $\tau=\{X,\emptyset,\{0\}\}$, the point $0\in\{0\}$ and $\{0\}\in X$. If I come up with other topology since I need to have $X$ in the topology there will be always two open sets containg the same point.

Question:

So how do I build a topology on this case that $(X,\tau)$ will be a $T_0$-space but not a $T_1$-space? What am I thinking wrong?

Thanks in advance!

Answer 1

I'm not sure what your doubt is. The family $\tau=\{\emptyset,\{0\},X\}$ is a topology on $X=\{0,1\}$.

It is $T_0$, because there is an open set containing $0$ and not $1$, precisely $\{0\}$. Since these are all the points in $X$, we are done.

On the other hand, $\{0\}$ is not closed, because the closed sets are $X$, $\{1\}$ and $\emptyset$. Thus the topology is not $T_1$.

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There are not many choices for a topology on $X=\{0,1\}$: the following is the complete list.

1. $\{\emptyset,X\}$, the trivial topology; not $T_0$.
2. $\{\emptyset,\{0\},X\}$; it is $T_0$ but not $T_1$.
3. $\{\emptyset,\{1\},X\}$; it is $T_0$ but not $T_1$.
4. $\{\emptyset,\{0\},\{1\},X\}$, the discrete topology; it is $T_2$.

Answer 2

$\tau =\{X,\{\},\{1\}\}$ or $\tau =\{X,\{\},\{0\}\}$

Answer 3

Not every singleton can be closed or $X$ would be $T_1$. So pick e.g. $\{1\}$ to not be closed. Then $\{0\}$ cannot be open, as its complement. This means the only open set that contains $0$ must be $\{0,1\} = X$. But $\{1\}$ must be open for $X$ to be $T_0$ (we must distinguish $1$ from $0$). $X$ and $\emptyset$ are always open.

So we get $\{\emptyset, X , \{1\}\}$ as the topology (we can also choose $0$ to be the open singleton, and we get a different, but essentially the same, topology). It's called the Sierpiński two-space.