Finding a Basis and Sub-Basis for a given set X. Suppose $X = \{a,b,c,d,e\}$
then suppose $\mathscr{B} = \{\{a\},\{a,c\},\{a,b\},\{b\},\{d\},\{e\},\{c\}\}$.
Is this a basis for $X$?
Part 1 of the definition states that for each $x \in X$ there is at least one basis element $B \in \mathscr{B}$ containing $x$. 
So far I've noticed that all the elements in $X$ are also in $\mathscr{B}$ and that there is a basis element $\{a\}$ and $\{a,c\}$.  
Now for Part 2, I found that taking the intersections of the basis elements $\{a\}$ and $\{a,c\}$ to get $\{a\}$. Then it applies for the third basis element which is a subset of the intersection of the two Basis elements.
Then $\mathscr{B}$ generates the topology $\mathscr{T}$
The next thing that I'm confused about is finding the Sub-Basis for the set $X$.
If someone can explain how to find the subbasis for $X$ by using an example that's similar to the one above, then that would be appreciated.
Is this a basis or simply something that isn't a basis?
 A: If you need a reference, I'll suggest Section 13 of the second edition of Topology by Munkres. A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$. For example, with your set $X$, a subbasis would be $\{ \{ a,b \}, \{a, c, d, e \} \}$. 
The topology generated by $S$ is then then the collection of all unions of finite intersections of elements of $S$. So in the example given above, the topology is $\{ \{\}, \{a\}, \{a,b\}, \{a,c,d,e\}, X \}$. You can check for yourself that this is indeed a topology. 
A subbasis and basis are distinct concepts, but the finite intersections of elements of the subbasis form a basis. Any basis is trivially a subbasis, but the converse need not be true (for example, the example I gave above is not a basis). 
A: In your example, the collection is a base for a topology on $X$. All singleton sets $\{x\}$ where $x \in X$ are in $\mathscr{B}$ (you say elements of $X$ but $x$ is not the same as the set $\{x\}$!). So indeed $X$ is the union of all sets in $\mathscr{B}$ which is condition (1) of being a base for some topology (and in some texts this suffices to show that it is at least a subbase for a topology).
To check condition (2), you need to do a bit more, but it follows from the singleton observation above: if $B_1, B_2$ are in $\mathscr{B}$ and we have some $x \in B_1 \cap B_2$, then $B_3 := \{x\} \in \mathscr{B}$ and so indeed we have $x \in B_3 \subseteq B_1 \cap B_2$, which is the other condition one has to check for being a base for a topology on $X$. 
Conclusion: $\mathscr{B}$ is a base for a topology (and all sets of $X$ are open in this topology, as every set $A \subseteq X$ can be written as $\cup_{x \in A} \{x\}$ and so is open). So $\mathscr{T} = \mathscr{P}(X)$, the discrete topology.
Any base is also a subbase, so $\mathscr{B}$ is a subbase for the same topology as well. But there are many more. E.g. $\mathscr{S} = \{ \{a,b\}, \{a,c\}, \{b,c\}, \{a,d\}, \{d,e\}, \{b,e\}\}$ is one as well (taking finite intersections we can get all singletons sets again and then taking unions we get all subsets of $X$ again); note that this is not a base. So the easy way out for the question of giving a subbase for $\mathscr{T}$ is just saying $\mathscr{B}$ again, but we can take $\mathscr{S}$ as well, or the set $\{\{a\}, \{b\},\{c\}, \{d\}, \{e\}\}$ as well (which is both a minimal base and a minimal subbase).
