Sub-bases for classical topological constructions

Question

I was wondering whether given a collection of topological spaces $\{(X_\alpha,\tau_\alpha)\}_{\alpha \in \Lambda}$ and sub-bases $\mathcal{S}_\alpha \subset\tau_\alpha$, can we obtain sub-bases from this sub-bases to classical constructions? I wonder whether these following attempts of giving a sub-base for these constructions is indeed true:

Sub-space topology:

Given a topological space $(X,\tau)$ a sub-base $\mathcal{S}=\{ S_i \}_{i\in I}$ and a subset $G\subset X$, a sub-base for the subspace topology is $\mathcal{S}_G=\{S_i\cap G\}_{i\in I}$.

Disjoint Union topology:

The disjoint union of $\{(X_\alpha, \tau_\alpha)\}_{\alpha\in \Lambda}$ with subases $\{ \mathcal{S}_\alpha \}_{\alpha\in \Lambda}$, has $\cup_{\alpha \in \Lambda} \mathcal{S}_\alpha$ a sub-base.

Box topology:

The box topology on $\prod_{\alpha\in \Lambda} X_\alpha$, has $\prod_{\alpha\in \Lambda}\mathcal{S}_\alpha$ as a sub-base.

Product topology

For all finite $F\subset \Lambda$, I can define $\mathcal{S}_F:=\{U=\prod_\alpha U_\alpha: U_\alpha \in \mathcal{S}_\alpha \; \text{when } \alpha\in F,\; U_\alpha=X_\alpha \; \text{if } \alpha\notin F \}$, and then $\mathcal{S}:=\cup_{F\subset \Lambda, \vert F\vert<\infty} \mathcal{S}_F$ is a sub-base for the product topology.

Quotient topology:

Given a topological space $(X,\tau)$ with sub-base $\mathcal{S}$ and an equivalence relation $\sim$, then the quotient space $(X/\sim,\tau_{\sim})$ has $\mathcal{S}_\sim:=\{ [U]_\sim :\; U\in \mathcal{S} \}$ a sub-base.

This seems true to me, but I have been wrong before so I wanted to verify whether this is true. I should also note that when I say sub-base, it does not have to be a cover.

Answer 1

Some of these constructions are classic examples of so-called initial topologies on which I wrote extensively in this answer before.

If $X$ has the initial topology w.r.t. maps $f_i: X \to (X_i, \tau_i), i \in I$ and we have subbases $\mathcal{S}_i$ for each $\tau_i$, then $$\{f_i^{-1}[S]: S \in \mathcal{S}_i\}$$ is a subbase for the initial topology on $X$ w.r.t. the $f_i$. This is an easy combination of the fact of the minimality of $\tau$ and of the fact that each $\tau_i$ is the minimal topology that contaisn $\mathcal{S}_i$, etc.

So for the subspace topology on $ \subseteq X$ this gives $\{S \cap G: S \in \mathcal{S}\}$, where $\mathcal{S}$ is a subbase on $X$, and for the product topology on $\prod_{\alpha \in \Lambda} X_\alpha$ we get the more economical subbase

$$\{\pi_\alpha^{-1}[S]: S \in \mathcal{S}_\alpha\}$$

where each such set only depends on one instead of finitely many coordinates.

There is (AFAIK) no such general fact for final topologies (of which quotient and sums are examples, see my post here e.g.) so there we have to judge case by case.

The box topology is neither initial nor final, but there the situation is indeed as you say: all sets of the form $\prod_{\alpha \in \Lambda} A_\alpha$, where each $A_\alpha$ is chosen from $\mathcal{S}_\alpha$ do form a subbase for the box topology. The proof of this is quite trivial, as products and intersections interact well.

For quotients your idea does not work, firstly because $[U]_{\sim}$ (if it is defined as $q[U]$ where $q$ is the canonical quotient map) need not be open (and this is a requirement), and because simple counterexamples exist: Take $X = \Bbb R$ (usual topology) and let $\sim$ be the equivalence relation with classes $\Bbb Z$ and $\{x\}, x \notin \Bbb Z$ (so we identify the integers to a point). If we take the countable base as a subbase of $X$ then its images cannot form a subbase for $X{/}{\sim}$ because the latter quotient space does not have a countable base (or subbase) at all (it's well-known that $q(0)$ does not have a countable local base, even).

For sums it's indeed as you state: if we make the necessary identifications so that each $X_\alpha$ is a real subset of $\oplus_{\alpha \in \Lambda} X_\alpha$, then $\bigcup_{\alpha \in \Lambda} \mathcal{S}_\alpha$ is a subbase for that sum, again (as for box products) as sums interact well with intersections.

So in short: your product subbase can be improved upon, the quotient one does not (and cannot) work, and the others are true and have easy proofs.