Lecture 1:
We know the definition of open sets. Closure under arbitrary unions and finite many intersections and empty set and X itself.
we have courser and finer. Finer is larger.
We have the finite complement topology.
The finite complement topology: Complements and $\emptyset$
X - $\bigcup U_\alpha$ = $\bigcap(X-U_\alpha)$
Let’s see if we can do this. $\implies$ It’s not in any $U_\alpha$ but in X. yeah good. $\impliedby$ i mean yeah, same notion.
Basis we know this. Intersections are basis. Definition: A topology generated by basis $\mathcal{B}$ for every x in the open sets, there is a basis completely contained in the open set and contains x.
For finite intersections, we have two bases, U1 U2 open in T. The intersection is open. Isn’t the rest here just trivial.
Lemma: Let B be a basis for a topology T on X.
Then, T is the collection of all unions of elements of B.
Proof: $B \subseteq T$ so arb unions of elements of B are in T.
Conversely, let $U \in T$. For each $x \in U$, there is a basis element such that $x \in B_x \in U$. Now, $U = \bigcup B_x$ is a union of elements of B. (basis elements)
okay, this is all easy I already did this.