Closure of Intersection may not equal Intersection of Closures/Examples
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Examples of Closure of Intersection may not equal Intersection of Closures
Arbitrary Subsets of $\R$
Let $H$ and $K$ be subsets of the set of real numbers $\R$ defined as:
\(\ds H\) | \(=\) | \(\ds \openint 0 2 \cup \openint 3 4\) | ||||||||||||
\(\ds K\) | \(=\) | \(\ds \openint 1 3\) |
Let $\map \cl H$ denote the closure of $H$.
Then:
- $H \cap \map \cl K$
- $\map \cl H \cap K$
- $\map \cl H \cap \map \cl K$
- $\map \cl {H \cap K}$
are all different.