62. Let R be the set of real numbers with the topology generated by the basis {[a,b):a<b, where a,b are in R}. If X is the subset [0,1] of R, which of the following must be true?

I. X is compact.

II. X is Hausdorff.

III. X is connected.

I don't get I . Isn't [0,1] closed and bounded so compact? For closedness: Since its complement (-infinity, 0) U (1,+infinity) is open in standard topology R, so it is also open in lower limit topology R_L, therefore itself, [0,1], is closed; and [0,1] is certainly bounded. What did I miss?

Thanks!