i need to mark statements below true or false and justify.(adsbygoogle = window.adsbygoogle || []).push({});

a) every nonempty finite set is compact.

since a finite set must have upper and lower bounds, thus it must be bounded. thus my question becomes that if a finite set is closed. i am considering a closed set [0,5] subset of R finite or not. it seems like it is not infinite set thus it is not compact. but i dont know if my arguement is valid or not.

b) no infinite set is compact.

false, counterexample is [0,inf) subset of R

c)the union of any collection of open sets is an open set.

true because it is the size of such set shall only increase and the boundry point of the union set never conatined in the set.

d)if a set has a max and a min, then it is compact.

max and min imply it is bounded and closed, thus it is a compact set.

e) if a nonempty set is compact, then it has a max and min.

similar arguement as d since the inverse is true.

and the last question was that find an example of a metric that is not shown in the book.

i use riemann integral[ |f-g|] and shown 4 properties of metric. what conditions do i need to put on functions f and g?

i said they are from R^n to R and continous. Do i need to state that they are increasing/decreasing monotonically too?

thx for any advice

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