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first it mentions in sec1-8 that a continuum product of sequencially compact spaces (therefore compact?) need not be sequentially compact (therefore not compact?)

then it proves thm1-28 that an arbitrary product of compact spaces in the Tychonoff topology is compact, the so called 'Tychonoff theorem'

then in an exercise it asks you to show that I^I is not compact in some unmentioned topology. isn't this an arbitrary product of compact spaces?

perhaps these are all distinct ideas, but its unclear to me what that is. i know whether or not the space is a metric space is an issue, but how?