Right now I'm getting into topology since I want to understand it more prior to fully learning Real Analysis.
I'm confused about how to construct both a Sub-Basis and a Base/Basis for a topology $X$.
I read the definition for Basis and Sub-Basis and find it a little confusing.
For example:
Definition. A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $T$ of all unions of finite intersections of elements of $S$.
So with this definition I'm confused about the second part where it describes the unions of finite intersections of elements of $S$. And what I'm grasping from the first part is that a subbasis contains elements that are in $X$ but there union creates the set $X$.
so If I have the set $X = \{a,b,c,d\}$
then $T = \{X,\varnothing, \{c,d\},\{b\}\}$ is a topology on $X$.
Then the subbasis $S = \{\{a\},\{b\},\{c\},\{d\}\}$.
I'm not sure if this is correct.
EDIT
Suppose $X = \{a,b,c,d,e\}$
then $B = \{\{a\},\{a,c\},\{a,b\}$,$\{b\},\{d\}$,$\{e\},${c}$\}.$
is a basis for X.
Is this a basis or simply nothing?
I'm currently using James Munkres Topology, 2nd Economy Edition.
Best Answer
I think it will help to understand exactly what your after when considering a subbase. Start with choosing a random collection of subsets $S$ of $X$. In general, you know that this collection $S$ does not form a topology. Now comes the fundamental question:
"What do we have to change about $S$ to get a topology?"
Well, for one, arbitrary unions and finite intersections of elements of $S$ ought to again be in $S$. If they are not, we simply add them to $S$ to get a new collection $T$.
Now we also need $X$ and the empty set $\emptyset$ to be in $T$ as well. However, if we add the condition that the union of elements of $S$ is $X$, we know that $X$ will be in $T$ by construction. This is what motivates the definition of a subbase.
In your example, $T=\{X,\emptyset, \{c,d \}, \{b\}\}$ is not a topology since the union $$\{c,d\}\cup \{b\}=\{c,b,d\}\not \in T.$$ Instead, let's consider $$T=\{\emptyset, \{b\}, \{c,d \}, \{c,b,d\}, X\}.$$ Does this form a topology? Check the axioms. Is your original example $$S=\{X,\emptyset, \{c,d \}, \{b\}\}$$ a subbase for T?