In the following, I am trying to consider the (sequential) compactness methods in applications from a rather abstract point of view. I am not sure, whether such an abstraction is really meaningful.
Preliminary remark: Let $X$ be a set, $\tau_1$ and $\tau_2$ two topologies on $X$ and consider the join topology $\tau_1 \vee \tau_2$, i.e. the topology generated by $\tau_1 \cup \tau_2$. If $x \in X$ and $x_\alpha$ is a net in $X$ that $\tau_1$-converges to $x$ and $\tau_2$-converges to $x$ then $x_\alpha$ $(\tau_1 \vee \tau_2)$-converges to $x$.
I often meet compactness theorems in applications that establish $\tau$-convergence of some sequence $x_n$ (for some topology $\tau$) by convergence in some weaker $\tau_1$-topology together with some additional properties that need to be satisfied by the sequence $x_n$. Sometimes, these additional properties can be rephrased by relative (sequential) compactness in another weaker topology $\tau_2$.
As an example, consider $X := L^1(P)$ for some probability measure $P$, $\tau$ the topology of convergence in $L^1$-norm, $\tau_1$ the topology of convergence in measure and $\tau_2 := \sigma(L^1, L^\infty)$ the weak topology. The Vitali convergence theorem states that on $L^1(P)$ a sequence $x_n$ $\tau$-converges if and only if $x_n$ $\tau_1$-converges and $x_n$ is uniformly integrable. By Dunford-Pettis and Eberlein-Smulian, the uniform integrability of $x_n$ is nothing else but relative sequential compactness in the $\tau_2$-topology. In particular, the Vitali convergence theorem implies that $\tau = \tau_1 \vee \tau_2$.
Question 1: Does it also hold the other way round? I.e. if $x_n$ is $\tau_2$-convergent and $\tau_1$-relatively sequentially compact does it follow that $x_n$ is $\tau$-convergent?
Question 2: If Question 1 has a positive answer, does it always hold that if on some (possibly nice enough) topological space $(X, \tau)$ it holds that $\tau = \tau_1 \vee \tau_2$ then $\tau_1$-convergence of a sequence or a net and $\tau_2$-relative (sequential) compactness implies its $\tau$-convergence? (Note that in contrast to the above preliminary remark, we do not impose the $\tau_2$-convergence (it will then follow automatically)).
Question 3: It would be also interesting to know of such "additional properties" that can not be rephrased by a relative (sequential) compactness property.