# Example of a topological space which is neither sequential nor $T_0$

What would be examples of topological spaces that are neither sequential nor $$T_0$$?

$$\pi$$-Base does not have an example of such space.

I have seen some examples of non-sequential spaces. Arens-Fort space is mentioned in this post. Another space mentioned there is the Stone-Čech compactification $$\beta\mathbb N$$. However, both of them are Hausdorff.

From non-Hausdorff spaces I was able to find the cocountable topology on an uncountable set. However, it is still a $$T_1$$-space.

So, the question remains of giving simple examples of non-sequential spaces which are not $$T_0$$.

• The reason for posting this was the delete-undelete war on this post. (See also meta.) This also means that I know the answer - I have seen it on the deleted post. The repeated deletion and undeletion suggests that some users consider this question useful and some do not. I hope that this version of the question has the better form than the original one. Perhaps in this way everybody is satisfied. Sep 2, 2016 at 21:32
• Can't you just pick a point $x_0$ that is in your favourite non-sequential space $X$ and a point $x_1$ that is not and then topologise the disjoint union $Y = X \sqcup \{x_1\}$, by taking $A\subseteq Y$ to be open if $A \cap X$ is open in $X$ and either (1) $x_0 \not\in A$ or (2) $\{x_0, x_1\} \subseteq A$? Then every open subset of $Y$ that contains $x_0$ also contains $x_1$, so $Y$ is not $T_0$, while a sequentially open subset of $X$ that is not open will still by sequentially open but not open in $Y$. Sep 2, 2016 at 22:10
• @RobArthan Yes, that seems like a rather natural way to construct such an example. In fact, this is precisely the space described in Brian M. Scott's answer in the paragraph about "a very simple machine". Sep 2, 2016 at 22:23
• Indeed: I was typing my comment while Brian was typing his answer, Sep 2, 2016 at 22:28
• @MartinSleziak Well played.
– user99914
Sep 3, 2016 at 2:41

For a very elementary example let $Y$ be an uncountable set, $y_0$ and $y_1$ distinct points of $y$, and $p$ a point not in $Y$. Let $X=Y\cup\{p\}$, $\mathscr{U}=\wp(Y\setminus\{y_0,y_1\})$, and $\mathscr{V}=\big\{U\cup\{y_0,y_1\}:U\in\mathscr{U}\big\}$. Finally, let

$$\tau=\mathscr{U}\cup\mathscr{V}\cup\{X\setminus C:C\in\mathscr{U}\cup\mathscr{V}\text{ and }C\text{ is countable}\}\;;$$

then $\tau$ is a topology on $X$. Every member of $\tau$ contains either both or neither of the points $y_0$ and $y_1$, so $\langle X,\tau\rangle$ is not $T_0$. $Y$ is sequentially closed, since the convergent sequences in $Y$ are those that are eventually constant or eventually in the set $\{y_0,y_1\}$, but $p\in\operatorname{cl}_XY$, so $Y$ is not closed in $X$.

One that fails to be sequential but still has non-trivial convergent sequences can be obtained by letting $Y=\omega_1+1$ with the order topology $\tau'$ and $p$ be a point not in $Y$. Let $X=Y\cup\{p\}$, and let

$$\tau=\{U\in\tau':0\notin U\}\cup\big\{U\cup\{p\}:0\in U\in\tau'\big\}\;$$

then $\langle X,\tau\rangle$ is not $T_0$, because every open set contains either both or neither of the points $0$ and $p$, and it’s not sequential, because $X\setminus\{\omega_1\}$ is a sequentially closed set that is not closed.

Each of these is the result of applying a very simple ‘machine’ to an arbitrary non-sequential space $\langle Y,\tau\rangle$. Fix some $y_0\in Y$, let $Z=\{0,1\}$ have the indiscete topology, and let $X=(Y\times Z)/{\sim}$, where $\langle y_1,z_0\rangle\sim\langle y_2,z_1\rangle$ if and only if either $\langle y_1,z_0\rangle=\langle y_2,z_1\rangle$, or $y_1=y_2\ne y_0$. This simply doubles the point $y_0$, turning it into two distinct points with identical nbhds in $X$, in such a way that $Y$ is the Kolmogorov quotient of $X$.

In fact it is true in general that if $X$ is non-sequential, then the Kolmogorov quotient $Y$ of $X$ is non-sequential, so every example of a non-sequential, non-$T_0$ space can be obtained by ‘fattening up’ a $T_0$ non-sequential space, albeit not necessarily at just one point.