Given, two compact sets A and B, how do we show that their intersection A$\displaystyle \bigcap$B is also compact?
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Originally Posted by harish21 Given, two compact sets A and B, how do we show that their intersection A$\displaystyle \bigcap$B is also compact? In what kind of space?
Originally Posted by Drexel28 In what kind of space? A and B are compact subsets of R
Originally Posted by harish21 A and B are compact subsets of R Think closed subspace
Originally Posted by Drexel28 Think closed subspace A and B are closed and bounded because they are compact. So A $\displaystyle \bigcap$ B is also closed. How do we show boundedness?
Originally Posted by harish21 A and B are closed and bounded because they are compact. So A $\displaystyle \bigcap$ B is also closed. How do we show boundedness? The Heine-Borel theorem is kind of overkill here...but. Clearly if $\displaystyle C\subseteq D$ then $\displaystyle \text{diam }C\leqslant \text{diam }D$, right? $\displaystyle A\cap B\subseteq A$
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