Containment relationships between elements of two arbitrary bases

Question

We have a topological space $X$ with bases $\mathcal{B_1}$ and $\mathcal{B_2}$. For $x\in X$, there is a $B_{1,x}\in\mathcal{B_1}$ and $B_{2,x}\in\mathcal{B_2}$ which contain $x$, by definition of basis.

Since it is a basis, without loss of generality $\mathcal{B_1}$ generates the topology $X$, and so for any open set $U$ in $X$ and $x\in U$ there is a basis element $B_{1,x}$ with $x\in B_{1,x}$ and $B_{1,x}\subseteq U$. Then it follows that such a basis element in $\mathcal{B_2}$, say $B_{2,x}$ contains such $B_{1,x}$, since $B_{2,x}$ is an open set; and likewise for $B_{1,x}$, which is open as well, so that there is a basis element $B'_{2,x}\subseteq B_{1,x}$, etc.

In this fashion, having fixed two bases of $X$, each point of $x$ "induces" a nested collection (or sequence, if it is countable) of 'alternating' basis elements, which may possibly 'terminate' if at some point along this nesting it holds that $B^\alpha_{1,x}=B^\beta_{2,x}$, where $\alpha,\beta$ are in some index sets which index the basis elements of $\mathcal{B_1}$ and $\mathcal{B_2}$, respectively, which participate in this nested collection.

Is this above correct?

Edit This may be a related question: Constructing a local nested base at a point in a first-countable space

Answer 1

You can try such a recursive construction, but it might very well fail at limit steps:

Suppose we have $x \in X$ and $B^1_0 \in \mathcal{B}_1$ that contains $x$. Then there is a $B^2_0 \in \mathcal{B}_2$ such that $x \in B^2_0 \subseteq B^1_0$, then we pick $B^1_1 \in \mathcal{B}_1$ such that $x \in B^1_1 \subseteq B^2_0$, etc.

So we try to construct $B^i_\alpha$, where $i=1,2$ and $\alpha$ an ordinal such that $\forall \alpha < \beta$: $x \in B^1_\beta \subseteq B^2_\alpha \subseteq B^1_\alpha$, and we can go on for $\alpha \in \omega$. and it can stabilise when we have some common $B \in \mathcal{B}_1 \cap \mathcal{B}_2$, containing $x$, that we can keep on choosing. In a regular space we could choose the closure of the base element to sit inside the prviosu one, and we stabilise at clopen common base elements, e.g. (we could have isolated points).

But at limit stages it's unclear what to do, we can only continue if $\cap_{\alpha<\beta} B^i_\alpha$ would have non-empty interior. But if a point is a $G_\delta$ we can easily get stuck at stage $\omega$ of a recursion, because then that intersection could be just $\{x\}$ very easily.

So you can make mutually nested decreasing base sequences, but even these might trivialise if you don't take precautions.

What do you want to achieve with these constuction ideas?