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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Check my work (Spivak problem in Calculus on Manifolds)

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