My (naive) question is about getting an understanding on weak/strong topologies.
I know that if we have two topologies in $X$, say $T_1$ and $T_2$, $T_1$ is weaker than $T_2$ when the open sets in $T_1$ are open sets in $T_2$. This is something that I conceptually understand. I have seen it symbolized as i) $T_1 \subseteq T_2$.
On the other hand I have seen the expression that $T_2$ is stronger than $T_1$, sometimes symbolized as ii) $T_2 \succ T_1$ iff $\forall B\in T_1 \exists A\in T_2 : A\subseteq B $.
Q1: Am I right about i) and ii) and are they the same thing?
Q2: If I have an open set in $T_1$ I am pretty sure we can say it's open in $T_2$. (correct?)
Q3: When we say that a function $f:X \to Y$ is continuous in the one or the other topology (lets say it's continuous in the strong),are there any default assumptions about continuity in the other?
(To give you some context, I am auditing for fun a functional analysis class and I reached the point where we talk about weak topologies that have to do with dual spaces, weak star topologies etc. I saw that:
norm topology $\succ$ strong topology $\succ$ weak topology $\succ$ weak star topology
So by understanding what happens with the two topologies I am hoping to understand the above).
Q4: I read multiple MathstackExchange older questions and came across the following expressions: "weak-weak" , "weak-strong", "strong-strong" etc. What do these mean? For example what the phrase "If $f$ is continuous "A-B" then it's continuous "W-Z"" means? (If $f:X \to Y$ then A and W are about X and B and Z are about Y ?) Could you direct somewhere to read about these things or give me a very laid out example?
Q5: What is the difference between the following expressions?
a) $f_n$ converges weakly to $f$
b) $f_n$ converges to $f in the weak topology
c) $f_n$ converges to $f$ in the weak (dual) topology
I thought b and c are the same (because the weak topology is the dual topology).
I could keep asking but I think I already asked too much, once I understand these I will create another post for the rest. Thank you in advance.