Closure of Intersection may not equal Intersection of Closures

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $H_1$ and $H_2$ be subsets of $S$.

Let ${H_1}^-$ and ${H_2}^-$ denote the closures of $H_1$ and $H_2$ respectively.


Then it is not necessarily the case that:

$\paren {H_1 \cap H_2}^- = {H_1}^- \cap {H_2}^-$


Proof

Note that from Closure of Intersection is Subset of Intersection of Closures it is always the case that:

$\paren {H_1 \cap H_2}^- \subseteq {H_1}^- \cap {H_2}^-$

It remains to be shown that it does not always happen that:

$\paren {H_1 \cap H_2}^- = {H_1}^- \cap {H_2}^-$


Proof by Counterexample:

Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.

Let $H_1 = \openint 0 {\dfrac 1 2}$ and $H_2 = \openint {\dfrac 1 2} 1$.

By inspection it can be seen that:

$H_1 \cap H_2 = \O$

Thus from Closure of Empty Set is Empty Set:

$\paren {H_1 \cap H_2}^- = \O$


From Closure of Open Real Interval is Closed Real Interval:

$H_1 = \closedint 0 {\dfrac 1 2}, H_2 = \closedint {\dfrac 1 2} 1$

Thus:

${H_1}^- \cap {H_2}^- = \set {\dfrac 1 2}$

So $\paren {H_1 \cap H_2}^- \ne {H_1}^- \cap {H_2}^-$

Hence the result.

$\blacksquare$


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