Im kinda new studying topology, and one of the concepts I find hard to fully understand are basis. From the definition $B$ is a basis if for each open set $O$ in the given topology and each point $x$ of it, there exists a $b$ subset of $B$, such that $x$ its contain in $b$ and as wall as in $O$.
The topology given is: Let $A\subset X$, and $\tau:=\{X\} \cup \{O \subset X : O \cap A = \emptyset\}$
I proceed to prove that $\tau$ is a topology.
It's clear that $X \in \tau $, and $\emptyset \in \tau$ since the intersection with is always empty.
Let, $O_1,O_2 \in \tau$ it implies that $ O_1\cap A = \emptyset$ and $O_2\cap A = \emptyset$ Now, to prove that their intersection is in $\tau$ then $ (O_1 \cap O_2) \cap A = \emptyset$ but by the distributivity of the intersection it follow $ (O_1 \cap O_2) \cap A = (O_1 \cap A) \cap (O_2 \cap A)$ which because they are open, it's empty, so $(O_1 \cap O_2) \in \tau$
To prove the final, let $\{O_n\}_{n\in I} \in \tau$, so $\forall n \in I$ $O_n \in \tau$, further more, $O_n \cap A =\emptyset$. To prove that $\bigcup_{n \in I} O_n \in \tau$, then $ \bigcup_{n \in I} O_n \cap A =\emptyset $ which is true because $ \bigcup_{n \in I} O_n \cap A = \bigcup_{n \in I} (O_n \cap A)$ and since for each $n$ that instersection is empty, so the union must be empty as well.
To find a basis (not trivial) I have to construct a family of sets such that, there is an open set that contains them.
I say that $B=\{u\subset X : u \subset A\}$ is a basis for $\tau$, it's the only collection that I can think off.
But to show that indeed is a basis I don't see clear that I can find such a set $b \in B$. Further more, that $\forall O \in \tau$, and $\forall x \in O$, $\exists b \in B$ such that $x\in b \subset U$. Even more, how to prove this.
Any help in constructing a basis and tips are welcome.
Thank you in advance.
Also, I'm truly sorry If there is a mistake in the spelling or the grammar.
That topology is called the exclude A topology.
The smallest base is { {x}, X : x not in A }.
Exercise. What is the closure of {a} if a in A?