Convergence in topologies

Let $\tau_1\subseteq \tau_2$ be two Hausdorff regular topologies in an infinite set $X$ such that the convergences of sequences in $\tau_1$ and $\tau_2$ coincide (they have the same convergent sequences (to the same limits)). Must the Borel $\sigma$-fields (generated by open sets) of $\tau_1$ and $\tau_2$ be the same?

Solution 1:

They need not be the same. Let $X = ^{\omega_1}\mathbb{Z}$, the set of integer sequences of length $\omega_1$, and give it the order topology $\tau_1$ induced by the lexicographic order. Every linearly ordered topological space is $T_1$ and hereditarily normal, so $\langle X,\tau_1 \rangle$ is certainly $T_3$. There are no non-trivial convergent sequences in $\langle X,\tau_1 \rangle$. To see this, suppose that $x \in X$, and $A \subseteq X\setminus\{x\}$ is countable. There is an $\alpha < \omega_1$ such that for each $y\in A$ there is some $\xi < \alpha$ such that $x(\xi)\ne y(\xi)$. Let $x^-,x^+\in X$ be be defined as follows: $$\begin{align*} x^-(\xi) &= \begin{cases} x(\xi),&\text{if }\xi \ne \alpha\\ x(\alpha)-1,&\text{if }\xi = \alpha \end{cases}\\ &\\&\\&\\ x^+(\xi) &= \begin{cases} x(\xi),&\text{if }\xi \ne \alpha\\ x(\alpha)+1,&\text{if }\xi = \alpha. \end{cases} \end{align*}$$

Then $x \in (x^-,x^+) \subseteq X\setminus A$.

Now let $\tau_2$ be the discrete topology on $X$; clearly $\tau_1 \subseteq \tau_2$, and there are no non-trivial convergent sequences in $\langle X,\tau_2 \rangle$, either. But the Borel $\sigma$-field of $\tau_2$ is $\wp(X)$, and I’m reasonably sure that that of $\tau_1$ isn’t.

Edit: I originally had $X = ^{\omega_1}2$ with the lexicographic order topology, which, as Byron Schmuland pointed out, does have convergent sequences. I was actually thinking of the tree topology on $X$, which has a base consisting of all sets of the form $\{x \in X:x \upharpoonright\alpha = \varphi\}$, where $\alpha < \omega_1$ and $\varphi \in ^{\alpha}2$. It’s zero-dimensional and $T_1$, hence completely regular, and every countable subset is easily seen to be closed and discrete by an argument similar to the one used above to show that $\langle X,\tau_1 \rangle$ has no non-trivial convergent sequences..