Arbitrary product compactness

[xaos]

i'm reading Hocking&Young(Dover), and its clear I've missed something in my understanding.

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?

 

[Tinyboss]

Tychonoff does show that any product of compact spaces is compact (a "product" of spaces always implies the product topology, by the way). So the I^I example must be using a different topology, maybe box.

Also, compact spaces are sequentially compact and limit point compact, but in general the converse doesn't hold. It does hold in metrizable spaces, where the three notions are equivalent.