Closure of Intersection may not equal Intersection of Closures/Examples/Arbitrary Subsets of Real Numbers

Examples of Closure of Intersection may not equal Intersection of Closures
Let $H$ and $K$ be subsets of the set of real numbers $\R$ defined as:

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.

Proof
From Closure of Open Real Interval is Closed Real Interval:

Hence by definition of set intersection:

All defined sets, as can be seen, are different.