Difference between a basis of a topology and a generator of a topology 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!
 A: 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:$$\tau=\left\{\bigcup\mathcal B_0\mid\mathcal B_0\subseteq\mathcal B\right\}$$
A: 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:

*

*Every open subset $ U $ of $ (X,\tau) $ can be written as a union of finite intersections of elements of $ \mathcal F $.

*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).
