Consider R with the Euclidean metric Let S be a compact subs

Consider R- with the Euclidean metric. Let S be a compact subset of R2. Consider the projection of S given by X(S) = {x1 epsilon R | such that (x1,x2) epsilon S}. Show that X(S) is a compact subset of R (with the Euclidean metric).

Solution

A subset X(S) belonging to the ser R will be called a compace subset of R if every open covering of X(S) contains a finite subcovering.

if we could prove that the subset X(S) is finite then we can say that its a compace subset.

we are given that S is a compace subset of R2 , so this implies that S is a finite subset.

and we are given the projection of S as

X(S) = {x1 E R | exists X2 such that (x1,x2) E S}

and from this projection and the fact that S is finite we can say that

X(S) will be a finite subset as well.       ------> (1)

and as x1 E R , ----------> (2)

using (1) and (2) we cn say that

X(S) is a compact subset of R.