build a topology with $ X = \{a,b,c,d,e,f \} $ I need to build a topology with $  X = \{a,b,c,d,e,f \} $ but I have very clear how? 
there are so many elements.
I know the properties of topological spaces, but not how 
anyone know how? 
is not well identify sets that can give me a topology.
I tried with:
$\{X,\{a\},\{b\},\{c\}\}$ and $\{X,\emptyset\}$ but I don't know.
What is the minimal topology of $X$?
 A: The properties of a topology $\tau$ on a space $X$ are as follows:


*

*$X\in \tau$ and $\emptyset \in \tau$ (the whole space and the empty set are open sets in $X$).

*If $U\in \tau$ and $V\in \tau$, then $U\cap V \in \tau$ (finite intersection of open sets are open).

*If $\mathcal F \subseteq \tau$, then $\bigcup_{U\in \mathcal F}U \in \tau$ (arbitrary unions of open sets are open).


In your first example you have $\{a\} \in \tau$ and $\{b\} \in \tau$, but their union, $\{a, b\}$ is not in $\tau$. therefore this cannot be a topology.
Your second example $\tau = \{X, \emptyset\}$ will be a topology for any space $X$. It is the simplest topology allowed, and it is therefore called the trivial topology.
The following topology also has a name: $\tau = \{U \mid U \subseteq X\}$, containing all subsets of $X$. It is called the discrete topology. Another valid topology is $\tau = \{\emptyset\} \cup \{U \mid a \in U\}$ where a set is open iff it contains $a$ or is the empty set.
All in all there are a lot of different topologies on your $X$. If you want to make your own example, do this: Start with a few subsets, containing $X$ and $\emptyset$ (this starting point is called a sub-basis for the resulting topology). Now add to your collection all possible intersections of your sets (only intersections between finitely many sets are allowed, but that makes little difference when the set itself is finite) (what you now have is called a basis for the topology you end up with). Then append all possible unions of those sets again. You now have a full-fledged topology on your space.
A: Here you have an example of a set that is not a topology when $X=\{a,b,c,d,e\}$
$$\mathcal{O}=\{X,\emptyset \{a\}\{c,d\}\{a,c,e\}\{b ,c,d\}\}$$
$\mathcal{O}$ is not a topology on $X$ because the union of $\{c,d\}\cup \{a,c,e\}=\{a,c,d,e\}$ does not belong to $\mathcal{O}$ 
Here you have some examples of topologies if $X=\{a,b,c,d,e\}$
Minimal Topology :
 $$\mathcal{T}_a=\{X,\emptyset , \{a\}\}$$
$$\mathcal{T}_b=\{X,\emptyset , \{b\}\}$$
$$\mathcal{T}_c=\{X,\emptyset , \{c\}\}$$
And 
$$\mathcal{T}=\{X,\emptyset , \{a\}\{c,d\}\{a,c,d\}\{b,c,d,e,f\}\}$$. Then  $\mathcal{T}$ is a topology because satisfies all the conditions of topology
