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In general topology the idea of a basis is quite simple. The definition is:

Let $X$ be a set, a set $B\subset \mathcal{P}(X)$ is said to be a basis for a topology on $X$ if:

  1. For each $x\in X$ there's $U\in B$ such that $x\in U$.
  2. If $x\in U_1\cap U_2$, with $U_1,U_2\in B$, then there is $U_3\in B$ such that $x\in U_3$ and $U_3\subset U_1\cap U_2$.

With that, the topology $\tau$ generated by $B$ is defined so that $U\in \tau$ if for each $x\in U$ there's $U_x\in B$ with $x\in U_x\subset U$. In other words, $\tau$ is the set of all unions of elements of $B$.

Then one proves that $\tau$ is indeed a topology. Obviously this is the natural extension of the open balls we use in metric spaces. It is quite simple to understand and to get some intuition about it.

The other definition, I simply can't get any intuition about is the idea of subbasis. The definition is:

Let $X$ be a set, a set $S\subset \mathcal{P}(X)$ is said to be a subbasis for a topology on $X$ if the union of all sets on $S$ equals $X$. In that case, the set

$$B = \left\{S_1\cap\dots\cap S_n : S_i\in S, n\in \mathbb{N}\right\},$$

is a basis for a topology $\tau$ in $X$. In other words $\tau$ is the set of all unions of all finite intersections of elements in $S$.

If on the one hand the idea of basis is quite intuitive and simple to understand based on the simple example of open balls, the idea of subbasis seems quite different.

I mean, I know it works. The proof that $\tau$ is a topology is quite simple. What is not simple is to understand the intuition.

In that case: what is the intuition about subbasis? Why would anyone consider the object defined that way? Why is it relevant and how can we understand it properly to have some intuition on when we need to use it?

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  • $\begingroup$ The subbasis more $X$ (if the subbasis dont cover the whole space) induces a topology. Just take all countable unions and finite intersections. In short: any collection of subsets is a subbasis for some topology. $\endgroup$ – Masacroso Nov 2 '16 at 15:47
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Some authors don't even require a subbasis to have union equal to all of $X$, i.e. a subbasis is just any subset $S \subseteq \mathcal{P}(X)$ whatsoever. Whichever approach is adopted, the idea is just to use $S$ to generate a topology $\tau$ which includes $S$, and as few additional open sets as possible. Since an arbitrary intersection of topologies is a topology, one way to get $\tau$ is to take $$\tau = \bigcap \{ \tau' : \tau' \text{ is a topology with } S \subseteq \tau' \}.$$ But, it turns out we can also obtain $\tau$ by writing down a basis for it. Namely, $$B = \left\{S_1\cap\dots\cap S_n : S_i\in S, n\in \mathbb{N}\right\}$$ can be checked to be a basis for the topology $\tau$ above.

Anyway, I agree it's natural to be a bit suspicious at first of the definition of a subbasis. It seems too loose of a concept to be good for anything, right? But the point is to think of this as being something more akin to... say a generating set for a group. Given any subset $S$ of a group $G$, we can define the smallest subgroup $\langle S \rangle$ containing $S$. This can also be written down explicitly as $\{ g_1 \cdots g_n : g_i \in S \text{ or } g_i^{-1} \in S\}$. But here we don't find it strange that no assumption was made about $S$, right? We have just used any old subset of $G$ to generate a smallest group.

The situation in topology is the same. It is just that, because we use the special terminology "subbasis", we initially suspect that subbases should be somehow "special". But, in fact, a subbasis is just any old collection of subsets which we use to generate a smallest topology.

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