A coarser topology of the topology with 'compact set equals sequentially compact set'

I study about weak and weak* topology in functional analysis.

By Eberlein-Smulian, every weakly compact set is weakly sequentially compact. How about weak* topology? I learned that $(B_{X^*},\omega^*)$($\omega^*$ means weak* topology.) is metrizable when $X$ is separable, so it is clearly true for $(B_{X^*},\omega^*)$, but I don't know the result for $(X^*,\omega^*)$.

On the other hand, does this hold about general topology? i.e., if $(X,\tau_1)$ is a topological space that $\{K\subset X:K$ is compact$\}$=$\{K\subset X:K$ is sequentially compact$\}$ and $(X,\tau_2)$ is a coarser topology than $\tau_1$, does the same hold for $(X,\tau_2)$? I think it is false but cannot find examples.

• The weak$^*$ topology on $X^*$ is metrizable if $X$ is separable, but not in general. Commented Jan 11, 2017 at 9:31
• I confused something about weak* topology. Thanks a lot!
– CSH
Commented Jan 11, 2017 at 9:33
• You're welcome. The last statement of my comment was incorrect. I confused the content of the Banach-Alouglu theorem to be about the weak topology. Nevertheless, it is still relevant. Commented Jan 11, 2017 at 9:37
• Ok! Have a good day
– CSH
Commented Jan 11, 2017 at 9:38
• Here is a related post. One can show that for $X=\ell_1(\Bbb R)$, $B_{X^*}$ is not weak* sequentially compact. Commented Jan 11, 2017 at 10:14

The weak* topology of $X^*$ is never sequential, unless $X$ is finite-dimensional. To see this , you may modify this proof.
• That post asks whether or not every weakly sequentially closed set is weakly closed. This post asks whether or not weak$^*$ sequential compactness is equivalent to weak$^*$ compactness, which is the case when $X$ reflexive. Commented Jan 12, 2017 at 20:46
• @Aweygan, there are two questions: the first one is about metrasibility/sequentiality of $X^*$ in the weak*-topology. Commented Jan 13, 2017 at 6:15