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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 12th 2017
1. What do you mean by “being compact is strictly stronger than the condition of beiing sequentially compact”? – For a general topological space both properties are independent, see Wikipedia for an example of a compact space that is not sequentially compact.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 13th 2017

Thanks for catching this. Fixed.

2. Added comment on generalization to first-countable spaces of one implication.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 14th 2017

Thanks. But let’s make that more explicit: countably compact metric spaces are equivalently compact metric spaces