Show by using $\textbf{induction}$ that the finite intersection of $n$ open sets is open.
I've barely started to learn topology and the concept of open and closed sets. I honestly don't even know where to begin here. Other than the fact that I know that one of the three conditions of the definition of a topology is that the intersection of subsets must be open. Therefore it makes sense to me that the intersection of $n$ open sets must also be open. I've also seen similar proofs via contradiction but not induction.I don't know how to prove that using induction. Any idea would be greatly appreciated.
This question is asked and answered by another user but via contradiction. Though it is the exact same question, it doesn't help my cause since I'm supposed to solve this via induction.
Ok, so you know that the intersection of two open sets sets is open. That is your base case, $n=2$.
Induction hypothesis: The intersection of any $n-1$ sets is an open set.
Now let us try to decide if $A_1 \cap A_2 \cap \dotsb \cap A_n$ is open. Note that $$A_1 \cap A_2 \cap \dotsb \cap A_n = (A_1 \cap A_2 \cap \dotsb \cap A_{n-1}) \cap A_n.$$ By induction, the set in the parenthesis is an open set (it is an intersection of $n-1$ sets. But then this is an intersection of two open sets, namely $(A_1 \cap A_2 \cap \dotsb \cap A_{n-1})$ and $A_n$. The base case now tells us that this is also open, so the intersection of any $n$ sets is open.
If you are not familiar with induction, you should really have a look on several examples from different areas of mathematics.