I am reading the proof from General Topology by John L. Kelley using Alexanders Subbasis Theorem. I have a few question which I will denote in bold.

Let where each is compact and has the product topology.

Let be the subbasis for the product topology consisting of all sets of the form , where is the projection to the a-th coordinate and is open in

By Alexander:

is compact if every subcollection of s.t. no finite subcollection of covers , does not cover .

For each , let be the collection of all open sets in s.t.

(Isan open cover of?)

Then no finite subcollection of covers

(is this by hypothesis? Ifis an open cover of, doesn't this contradict the factis compact?)then by compactness of , a point for all in

(Why does compactness guarantee this point?)

The point whose a-th coordinate is belongs to no member of

is not a cover of .

Thank you for your help. I hope I worded my questions clearly enough