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Can someone explain the difference between a basis of a topology and a generator of a topology?

I already know that by taking unions of elements in the basis, you can construct the whole topology.

From the lectures I know that a topology generated by a set A, which is a collection of subsets of the topological space $(X,\mathcal{T}_X)$, is defined as the intersection of all topologies on $X$ containing $A$. So, $\mathcal{T}_X$ is the smallest topology on $X$, containing $A$.

Is it also true that you can take unions of elements in $A$ to construct the whole topology? I don't think that this is true. Take for example $X=\{1,2,3\}$ and $A=\{\{1\},\{1,2\}\}\subset \mathcal{P}(X)$. Right now you cannot construct $X$.

Thanks in advance!

Sam
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  • The topology generated by $A$ will in general include much in addition to the unions of elements of $A$; it also needs to include $X$, $\varnothing$, finite intersections, etc. – Arturo Magidin Jan 10 '23 at 15:10
  • @ArturoMagidin can you give an example? – Sam Jan 10 '23 at 15:40
  • You gave one: if $A={{1},{1,2}}$, then the collection of unions of elements of $A$ is just ${\varnothing,{1},{1,2}}$. It does not include the set $X$. If you start with $A={{1,2},{2,3}}$ (with the same $X$), the collection of unions of elements of $A$ is ${\varnothing, {1,2},{2,3},{1,2,3}}$, but this is not a topology, since ${1,2}\cap{2,3}={2}$ is not one of the sets, but the intersection of two open sets must be open. – Arturo Magidin Jan 10 '23 at 15:43
  • @ArturoMagidin can $A$ be an arbitrary collection of subsets? – Sam Jan 10 '23 at 16:15
  • basis is a generator but not conversely? – BCLC Jan 10 '23 at 16:31
  • @Sam; Yes, given any collection of subsets $A$, you can define "the least topology that contains $A$", by letting $\tau(A)= \cap{ \tau\subseteq\mathcal{P}(X)\mid\tau\text{ is a topology on }X}$. It may be that $\tau(A)$ is the discrete topology, though. – Arturo Magidin Jan 10 '23 at 17:37

2 Answers2

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For collection $\mathcal A\subseteq\mathcal P(X)$ define a collection $\mathcal B$ by stating that $B\in\mathcal B$ iff it can be written as a finite intersection of elements of $\mathcal A$.

So actually:$$\mathcal B=\left\{\bigcap\mathcal A_0\mid\mathcal A_0\subseteq\mathcal A\text{ and }\mathcal A_0\text{ is finite}\right\}$$

We allow $\mathcal A_0=\varnothing$ and respect the convention $\bigcap\varnothing=X$ so that $X\in\mathcal B$.

Then the topology generated by $\mathcal A$ is exactly the same as the topology generated by $\mathcal B$ and moreover $\mathcal B$ is a basis of this topology which means that the elements of this topology are exactly the arbitrary unions of sets in $\mathcal B$.

So if $\tau$ denotes this topology then:$$\tau=\left\{\bigcup\mathcal B_0\mid\mathcal B_0\subseteq\mathcal B\right\}$$

drhab
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This should have been a comment, but I've ended up being too wordy.

I think that the term "basis" is misleading here. Take a vector space $ V $. A basis of $ V $ is a collection $ \{e_1,\dots,e_n\} $ of (linearly independent) vectors $ e_j\in V $ such that the smallest vector subspace contained in $ V $ that contains the $ e_j $s is $ V $ itself.

The corresponding concept in topology is a subbase. Making up a proof to the following fact will make your ideas more clear. Let $ (X,\tau) $ be a topological space, and let $ \mathcal F $ be a collection of open subsets of $ (X,\tau) $. The following are equivalent:

  1. Every open subset $ U $ of $ (X,\tau) $ can be written as a union of finite intersections of elements of $ \mathcal F $.
  2. The topology $ \tau $ is the smallest topology on the set $ X $ such that contains every $ F\in \mathcal F $.

Typically, a subbase for a topological space is defined as in 1., but I prefer the intrinsic definition 2.

If you switch from topology to measure theory, you will discover that a "generator" for a measurable space as defined exactly as in 2, and that an explicit characterization of a generator in terms of its elements does not exists (Ok maybe there is one, but I forgot my descriptive set theory book at home and can't recall the details).

giovanniadeodato
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    This answer shows how to construct the smallest $\sigma$-algebra containing a collection of set $A$. The issue is that you have to go transfinite in that case, whereas with a topology sub-basis you don't. But the term "basis of a topology" is pretty standard in topology, and the fact that it is used differently elsewhere is not any more "misleading" than the fact that "normal" means very different things in analysis, probability, and in group theory. – Arturo Magidin Jan 10 '23 at 16:02
  • I don't want to change standard terminology, but I found that in most textbook (at least, in the textbooks popular in my country), (explicit descriptions of) subbases are introduced quite later. I found this confusing when I was learning the basic topological notions for the first time. Btw, feel free to edit my post if you think that "misleading" here is... well, misleading. English is not my native language so maybe I messed up smth – giovanniadeodato Jan 10 '23 at 16:07