# Weaker and Stronger topologies (open sets, continuity and terminology)

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.

For your first question, yes, they're the same. One can place a partial ordering on the set of all possible topologies on a space. The answer to your second question is also yes. If $$T_1$$ is weaker than $$T_2$$, i.e. $$T_1\subset T_2,$$ then every element of $$T_1$$ is an element of $$T_2.$$ Since the elements are open sets, this means that an open set in the $$T_1$$ topology is also an open set in the $$T_2$$ topology. Your second definition does the comparison using bases, instead, which is fine.
Your third and fourth questions are related. When we talk about continuous functions, a general definition is that $$f:X\rightarrow Y$$ is continuous if $$f^{-1}(U)$$ is an open subset of $$X$$ for every open subset $$U$$ of $$Y$$. So, in order to discuss continuity, one needs to define the appropriate notion of openness. That is, we need to know what the topologies are on $$X$$ and $$Y$$. For example, if we say that $$f:X\rightarrow Y$$ is weak-weak continuous, then we mean that it's continuous when $$X$$ and $$Y$$ are endowed with their weak topologies. Since we can compare various topologies, we can make statements on when continuity in one topology implies continuity in another. You can also do this using convergence, but you may need to use nets.
For your last question, I'd personally say that b) means weak convergence, c) means weak$$^*$$ convergence, and a) could be either of the above and needs more context to be able to tell. Sometimes, people will say "converges weakly," and they mean either weak or weak$$^*$$ convergence.