Problem 1322 (difficulty: 5/10)

Let \(\displaystyle H\subset\R^p\) be a bounded set. Are the following statements true or false?

(a) If \(\displaystyle \cl H\in\J\), then \(\displaystyle H\in\J\).

(b) If \(\displaystyle H\) is closed and \(\displaystyle H\in\J\), then \(\displaystyle \INT H\in\J\).

(c) If \(\displaystyle H\) is open and \(\displaystyle H\in\J\), then \(\displaystyle \cl H\in\J\).

(d) If \(\displaystyle k(\INT H)=b(\cl H)\), then \(\displaystyle H\in\J\).

(e) \(\displaystyle \partial H\in\J\).

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