Need help with understanding the Basis for a Topology

I don’t have a particular problem to share but I’m asking for assistance in describing a Basis or Sub-Basis for some Topology T.

So, if I have some set, say $\{a,b,c\}$ or $\{a,b\}$ what will the basis and sub-basis be?
I understand the definition of a Topology and how to find a topology for a given set $X$ but I’m having a hard time grasping the definition of Basis and Sub-basis for a topology.

I understand what the definition is saying but I’m having a really difficult time in seeing the visual representation of the definition. i.e. how to construct a basis with a given topology $T$ and set $X$.

All I’m looking for is some examples that gives me a good visual on how to construct any basis from some set $X$ and topology $T$.

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I think there are two things going on here: given any point set, you can construct a collection of subsets which satisfies the definition for a basis, and this in turn generates a topology by just taking unions of these sets. On the other hand, you can have a set with a topology already given, and have to determine a subset of this topology which satisfies the axioms for a basis. The first case I think is clear: you just find subsets, take their unions, and this is a topology. If you want your topology to have certain properties (Hausdorffness, second countable, etc.), you just pick your basis elements specifically so that these axioms hold. Finding a basis given a topology is more difficult.

Although it is kind of a special example, take $\mathbb{R}^n$. Given any open set (euclidean) and an element $x\in U$, we just want to find a family of open sets $\{B_{\alpha}\}$ such that there is $\gamma$ such that $x\in B_{\gamma}\subset U$. In fact, I think there’s a theorem in Munkres Topology section 13 which gives this result in more concrete language. Obviously, our example (or any metrizable space) has a given basis of open balls, and it is trivial to see that these actually satisfy the basis axioms (a proof of this is in chapter 2 of Rudin’s Principles of Mathematical Analysis).

For more exotic topological spaces, the question is a bit more difficult, but certainly not impossible.

Let’s take $X=\{a,b,c\}$. It’s a problem in Munkres that this has $29$ topologies.
The most obvious two bases we can think up are $\mathcal{P}(X)$ (which is just the discrete topology) and just $X$ (which is just the indiscrete topology). Now suppose we want this set to be Hausdorff: First, each point must be closed, so our basis ought to include $\{b,c\},\{a,c\},\{b,c\}$. But then the axioms for bases requires that $\{b,c\}\cap\{a,c\}=\{c\}$ has to include a basis element. So $\{c\}$ must be a basis element. Similar reasoning shows $\{a\},\{b\}$ are also open, so we have shown that any basis with the points closed on this topology must be discrete. With that, let’s just start by picking sets we want to be open. A basis has to cover the space. One example might be $\{a\}, \{b,c\}$. This is a good basis, and in fact generates the topology consisting of these two sets and $X,\emptyset$. Now, if we want $\{a,c\},\{b,c\}$ to be open, then we require that $\{c\}=\{a,c\}\cap \{b,c\}$ must be open, so this space generates the topology $\{\emptyset,\{c\},\{a,c\},\{b,c\},X\}$. We can keep going on by this: the idea is that we pick sets we’d like to be open, add sets until they cover the space, and then add sets such that for all $x\in B_1\cap B_2$ basis elements, there is basis element $x\in B_3\subset B_1\cap B_2$.

For a subbasis: the procedure is similar, except we only take finite intersections of sets, and add the set $X$ manually to gives a basis for a topology. So, really the above discussion also works for subbasis, except we don’t need to worry about covering the space. Some examples of subbases on $X=\{a,b,c\}$ are $\{x\}$ for $x=a,b,c$, $\{x,y\}$ for $x\neq y=a,b,c$, and etc. So you see that in practice what we’re doing when we’re building a basis is that we’re really building a subbasis by just ensuring we’re closed under finite intersections, since we can always add the total space $X$ to cover the space. In fact, although you can determine a subbasis from a topology, I think it is much more common that we take a collection of sets we’d like to be open and make them into a subbasis, which in turn determines a topology for a space, whereas determining a basis is usually done after we already know what our topology is, so we can reduce topological proofs to proofs with basis sets which are more “well behaved” than a full topology.

Let’s apply every above to the set $Y=\{a,b\}$. We want $\{a\}$ to be open? Then $\{a\},Y$ is a basis for $Y$, and this basis determines the Sierpenski space. In fact, this is really the only example: there are only $4$ topologies on this space, and two of them are essentially the same (in the above subbasis, just choose $\{b\}$ to be the set we want to be open).