# Difference between basis and subbasis in a topology?

I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?

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Bases and subbases "generate" a topology in different ways. Every open set is a union of basis elements. Every open set is a union of finite intersections of subbasis elements.

For this reason, we can take a smaller set as our subbasis, and that sometimes makes proving things about the topology easier. We get to use a smaller set for our proof, but we pay for it; with a subbasis we need to worry about finite intersections, whereas we did not have to worry about that in the case of a basis.

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Consider $S=\{\{0,1\},\{0,2\}\}$. What is the topological space $T(S)$ generated by $S?$ By definition, $S$ will then be a subbasis of $T(S)$.

Well, we want to all requirements to hold true and find that $T(S) = \{\emptyset, \{0\}, \{0,1\}, \{0,2\}, \{0,1,2\}\}$ (check this!).

Is $S$ a basis? No, because you cannot write $\{0\}$ as a union of any elements in $S$.

So you see that subbasis and basis are two different notions, even for a very basic example.

A subbasis can be thought of, and is actually defined to be, the "smallest set that becomes my topological space if I complete it under the property of being a topological space, i.e. fulfiling the axioms of topological space".

The two terms are related nevertheless. Every basis is a subbasis, and in one of the equivalent definitions of subbasis you will find that you already get a basis from your subbasis.

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Shouldn't a subbasis be able to produce every element of the topology by building the union of intersections of elements of the subbasis, where the intersections have to be finite? Something like: Let $(X, T)$ be the topological space and $S$ be a subbasis. Then: $\forall U \in T: U = \cup_{A = M_i \cap M_j \cap ... with M_{i,j} \in S}$ where all $A$ should be finite. How do you make the empty set this way? –  moose Feb 4 at 17:04
@moose You can make the empty set with the empty union of empty intersection. In other words the union of intersection of no sets. Kind of like how the vector space $\{0\}$ is generated by the basis $\varnothing$ since $0$ is an empty sum. –  Pratyush Sarkar May 24 at 23:21

The collection of sets $(-\infty,b)$ and $(a,\infty)$ for $a,b\in \mathbb{R}$ constitute a sub-basis for the standard topology on $\mathbb{R}$. The collection of sets $(a,b)$ for $a,b\in \mathbb{R}$ constitute a basis for the standard topology on $\mathbb{R}$. I suggest that you look at the definitions of "basis" and "sub-basis" and convince yourself that the claims that I have made are correct; this is probably the best way to answer your question.

The basic idea is that a basis is the collection of all finite intersections of sub-basis elements. The open sets in a topology are all possible unions of basis elements. So, the open sets in a topology are all possible unions of finite intersections of sub-basis elements.

If we ignore, momentarily, the fact that we are trying to generate a topology, a subbasis is any old collection of subsets of the space. Thus, any basis is a subbasis. However, a basis $\mathcal B$ must satisfy the criterion that if $U,V\in\mathcal B$ and $x$ is an arbitrary point in both $U$ and $V$, then there is some $W$ belonging to $\mathcal B$ such that $x\in W\subseteq U\cap V$.
To really understand what that means you have to look at examples. The family $\{X\}$ is a basis for the indiscrete topology on $X$. The family $\mathcal P(X)$, the powerset of $X$, is a basis for the discrete topology on $X$. If $X$ is large enough and you remove some subset $S\subset X$ from $\mathcal P(X)$, you can still have a basis if $S$ is not singleton but $\mathcal P(X)\setminus \{x\}$ is merely a subbasis when $x\in X$.
A more ambitious example is the definition of the topology on the product of two spaces $X$ and $Y$. The family $$\{U\times Y:U\subseteq X, U\text{ is open}\}\cup\{X\times V:V\subseteq Y,V\text{ is open}\}$$ is a subbasis but not necessarily a basis of the box topology on $X\times Y$.
Finally, suppose a family $\mathcal B$ is a basis. You can generate a topology $\tau_0$ using $\mathcal B$ as a subbasis and you can generate a topology $\tau_1$ using $\mathcal B$ as a basis. The topologies are actually the same, as you can see here. You just have to do more work to generate a topology from a subbasis $\mathcal S$, i.e. you have to include all finite intersections of members of $\mathcal S$ because it is not guaranteed that they are unions of members of $\mathcal S$.