Problem: given compact set C and open set U with C [tex]\subset[/tex]U, show there is a compact set D [tex]\subset[/tex] U with C [tex]\subset[/tex] interior of D. My thinking: Since C is compact it is closed, and U-C is open. Since U is an open cover of C there is a finite collection D of finite open subsets of U that cover C. Some of these subsets may be interior to C. Each of the other subsets contain a point x in (U-C), and since U-C is open, there is an open rectangle about x in U-C. Close these subsets at x. Then D is closed and bounded, D is a subset of U and C is interior to D. Are these statements true? Working alone I don't have anyone to check my reasoning. How does one write this with acceptable conventional notation? Thanks.