Closure of Intersection may not equal Intersection of Closures/Examples

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.