Topology for convergent sequences

Question

Let $(X,\tau)$ be a topological space, and consider the family $\mathcal{F}$ of the topologies over $X$ such that the convergent sequences for each $\gamma \in \mathcal{F}$ are the same as the convergent sequences for $\tau$, with the same limits. That is, $$ \mathcal{F} = \{\gamma \;|\; x_n \xrightarrow{\gamma} x \Leftrightarrow x_n \xrightarrow{\tau} x\} $$

It is easy to see that the topology $\tau_M$ generated by the topologies in $\mathcal{F}$ is such that $\tau_M \in \mathcal{F}$. The $\tau_M$ is the finest topology with the same convergent sequences as $\tau$, converging to the same point.

But is it true that the topology $$ \tau_m = \bigcap_{\gamma \in \mathcal{F}} \gamma $$ is itself in $\mathcal{F}$?

In other words, does it follow that the convergent sequences for $\tau_m$ and their limits are the same as the convergent sequences and their limits in $\tau$? Are there any nice counterexamples?

In other words again, do we have a weakest topology such that the convergent sequences and their limits are the same as the ones for $\tau$?

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Edit: Fixed according to @joriki's comment.

Answer 1

Taking the intersection can introduce new convergent sequences.

Let $p$ be a point not in $\omega$. Let $X=\{p\}\cup\omega$. For each free ultrafilter $\mathscr{U}$ on $\omega$ define a topology $\tau_{\mathscr{U}}$ on $X$ as follows. Points of $\omega$ are isolated, and the basic open nbhds of $p$ in $\tau_{\mathscr{U}}$ are the sets of the form $\{p\}\cup U$ such that $U\in\mathscr{U}$. The only convergent sequences of $\langle X,\tau_{\mathscr{U}}\rangle$ are the trivial ones. Let $$\tau=\bigcap_{\mathscr{U}\in\beta\omega\setminus\omega}\tau_{\mathscr{U}}\;.$$ A set $W\subseteq X$ belongs to $\tau$ iff $p\notin W$, or $W\in\mathscr{U}$ for all $\mathscr{U}\in\beta\omega\setminus\omega$. But the only subsets of $\omega$ that belong to all free ultrafilters on $\omega$ are the cofinite sets, and It follows that

$$\tau=\wp(\omega)\cup\{W\subseteq X:p\in W\text{ and }|\omega\setminus W|<\omega\}\;.$$

That is, points of $\omega$ are isolated in $\langle X,\tau\rangle$, and the basic open nbhds of $p$ are the sets of the form $\{p\}\cup(\omega\setminus F)$ such that $F$ is finite. But then $\langle k:k\in\omega\rangle$ is a sequence in $X$ that $\tau$-converges to $p$.

Now let $\tau'$ be the intersection of all topologies on $X$ having only the trivial convergent sequences. Clearly $\tau'\subseteq\tau$, so $\langle k:k\in\omega\rangle$ converges to $p$ in $\langle X,\tau'\rangle$.