Dealing with Tychonoff's Theorem.

Here are my few questions that I encountered while going through Tychonoff's theorem in .

a) First of all, so far I was thinking that Heine Borel definition of compactness implies sequential compactness but not the other way around ( although i am failing to find some examples to appreciate it). But what wikipedia says is that " but NEITHER implies the other in general topological space . What am i missing here ?

b) It is easy to see that finite product ( countable product is not true, right ? ) of sequentially compact spaces is compact which we can see using diagonalization argument . and it discusses of embedding X ( completely regular Hausdorff space ) into $[0,1]^{C(X,[0,1])}$ (what does $[0,1]^{C(X,[0,1])}$ mean? I am not able to make any sense) , where $C(X,[0,1])$ is the set of continuous map from $X$ to $[0,1]$. I would appreciate your help.
Thanks!

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2022-07-25 17:47:13
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